PHYSICAL REVIEW ACCELERATORS AND BEAMS 24, 063401 (2021)

Versatile, high brightness, cryogenic photoinjector electron source

River R. Robles ,*,† Obed Camacho, Atsushi Fukasawa ,Nathan Majernik , and James B. Rosenzweig

Department of Physics and Astronomy, University of California,Los Angeles, 405 Hilgard Avenue, Los Angeles, California 90095, USA

(Received 16 March 2021; accepted 25 May 2021; published 4 June 2021)

Since the introduction of the radio-frequency (rf) photoinjector electron source over thirty years ago,peak performance demands have dictated the use of high accelerating electric fields. With recent strongadvances in obtainable field values, attendant increases in beam brightness are expected to be dramatic. Inthis article, we examine the implementation of very high gradient acceleration in a high frequency,cryogenic rf photoinjector. We discuss in detail the effects of introducing, through an optimized rf cavityshape, rich spatial harmonic content in the accelerating modes in this device. Higher spatial harmonics giveuseful, enhanced linear focusing effects, as well as potentially deleterious nonlinear transverse forces. Theyalso serve to strongly increase the ratio of average accelerating field to peak surface field, thus aiding inmanaging power and dark current-related challenges. We investigate two scenarios which are aimed atunique exploitation of the capabilities of this source. First, we investigate the obtaining of extremely highsix-dimensional brightness for advanced free-electron laser applications. We also examine the use of amagnetized photocathode in the device for producing unprecedented low asymmetric emittance, high-current electron beams that reach linear collider-compatible performance. As both of the scenarios demandan advanced, compact solenoid design, we describe a novel cryogenic solenoid system. With the high fieldrf and magnetostatic structures introduced, we analyze the collective beam dynamics in these systemsthrough theory and multiparticle simulations, including a particular emphasis on granularity effectsassociated with microscopic Coulomb interactions.

DOI: 10.1103/PhysRevAccelBeams.24.063401

I. INTRODUCTION

The radio-frequency (rf) photoinjector is a class ofelectron source that has transformed numerous fields inbeam-based science, providing relativistic electron beamswith unprecedented short pulse length, high current, andlow emittance. This is accomplished by laser-gated photo-emission from a cathode embedded in a very high field rfcavity, liberating picosecond or faster electron pulsesthrough a prompt (as low as tens of femtosecond delay)photoelectric emission. It is after emission that the trueinnovation of the photoinjector begins, however, as if thebeam was simply accelerated from the cathode with noadditional optics it would experience strong correlatedemittance growth due to current-dependent transversespace-charge forces. However, it was shown by Carlsten

[1] that such correlated emittance growth could be reversedby focusing the beam soon after it emerges from thecathode, and one can in fact retrieve the emittance thebeam was born with. This concept was developed furtherby Serafini and Rosenzweig [2], where the process ofundoing current-dependent correlations, deemed emittancecompensation, became a prescribed process with well-understood working points to seek out for optimal com-pensation. The importance of these developments isstraightforward to appreciate—they enabled the robustperformance of the world’s first x-ray free-electron laser(XFEL) [3].The first generation of photoinjectors used in large

facilities provided beams which were of sufficient qualityto enable the world’s first XFEL. Today, however, we findmany innovative new applications for electron beamswhich demand ever-brighter sources. Unsurprisingly, theseinnovations in the use of high-brightness beams present aconcomitant challenge, to strongly increase the beambrightness produced by the source. In regard to XFELapplications, two recent initiatives indicate a necessity forbeams which far exceed the current state-of-the-art: theultracompact XFEL [4] and the MaRIE XFEL [5]. Thefirst, pioneered by a UCLA-centered collaboration, is anultracompact XFEL (UC-XFEL), which promises lasing,

*[email protected]†Present address: Department of Applied Physics, Stanford

University, Stanford, California 94305, USA.

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initially at soft x-ray wavelengths, with a total footprintunder 40 m in length. The fundamental feature allowingthe substantially shorter total length is the reduction of thefinal beam energy, from the several GeV level down to1 GeV. Doing so without dramatically reducing the FELpower requires maintaining the geometric emittance of thebeam at the lower energy, thereby demanding a dramaticreduction of the normalized emittance at emission. Furtherinsights into the demands on the electron beam brightnessresult from the analysis presented in [4], where it is shownthat the key performance metric for source improvement isthe six-dimensional brightness, which also includes thebeam spectral density.The second emergent FEL application, the matter

radiation interactions in extremes (MaRIE) XFEL, seeksto provide a photon source at extremely high photonenergies exceeding 40 keV, or equivalently an x-raywavelength below one-third of an Angstrom. Due to thestringent constraints imposed on beam emittance by theFEL instability, early design work toward such an FEL hasstruggled to identify a robust scenario, primarily due todegradation to beam brightness during transport to ener-gies in excess of 10 GeV. Even designs which utilize200 nm rad normalized emittance, at the current state-of-the-art of frontier injectors, meet performance metrics butwith uncomfortably little room for error. As such, thisproject would benefit greatly from an enhancement of thebeam brightness at the source, as was demonstrated by thework reported in Ref. [5].Like the x-ray free-electron laser, conceptual designs for

a future electron-positron linear collider demand extremelyhigh-brightness beams with even higher charge [6]. Thebeams required for a linear collider, however, go furtherthan requiring innovative approaches to high-brightnessbeam development. In order to mitigate beam-beam radi-ation effects at the interaction point, known as beams-strahlung, one wishes for the beams to be transverselyasymmetric, or “flat,” and in particular have asymmetrictransverse emittances [7]. Although there are severalapproaches to generating such beams, the only one capableof maintaining high brightness and high charge withoutexcessive additions to footprint or cost is the photoinjectoroperated with a magnetized photocathode. This entailsimmersing the cathode in an axial magnetic field such thatthe electron beam is born with a nonzero canonical angularmomentum which, as a conserved quantity, is converted tomechanical angular momentum downstream of thesolenoid region. The presence of this mechanical angularmomentum enables the splitting of the emittances via use ofa skew quadrupole triplet [8]. In the case of the linearcollider, we must maintain an ultrahigh brightness beamwhile purposefully immersing the cathode in a magneticfield—a situation which is traditionally avoided (e.g., inFEL applications) due to its potentially damaging effects onthe beam emittance.

Flat beams also find an application in dielectric laseracceleration (DLA) [9], particularly in structures with slabgeometries, where, in order to generate extremely highaccelerating gradients, the structures necessarily have gapswhich are very small in one transverse dimension, and verylarge in the other [10,11]. The use of an unmagnetizedphotoinjector to produce these beams is hindered by thesingle-nm emittance requirements in the small dimension.However, if the emittance is split then the smaller trans-verse dimension can achieve single-nm level emittancewithout requiring the same level thermal emittance at thephotocathode.The proposal that the path toward ever-brighter beams

lies in cryogenically cooled normal conducting rf structuresis at this point well established in theory and simulation, aswell as fundamental work on rf cavity performance [12].Previous work on the subject [13,14] has demonstrated thisin the simplest terms: with a standard sinusoidal accelerat-ing wave at the gun with 240 MV=m peak field, the beamsthat can be produced exceed the state-of-the-art by an orderof magnitude in brightness. Beams with these performancemetrics—55 nm rad normalized emittance, 20 A current,and sub-keV energy spread—have already been used insimulation studies of the UC-XFEL and MaRIE XFELdemonstrating excellent performance. However, new devel-opments in the burgeoning field of cryo-rf acceleratingstructures demand an updated treatment of such an injector.These new developments revolve largely around advancesin the design of copper accelerating structures, both that ofthe accelerating cavities themselves as well as the systemsused to couple rf power into them. The distributed couplinglinac [15] is unique among photoinjectors and acceleratingstructures alike in its square-wave-like field profile. Thenonsinusoidal profile is indicative of the presence of higherharmonic content in the accelerating wave, the effects ofwhich have been studied broadly in early studies of rfphotoinjectors but have never been studied in the context offields as high as we consider here, nor with beams as brightas those we present.The specific design requirements of the UC-XFEL have

also recently become mature. Reference [4] presents adetailed summary of these requirements, which elucidatednot just the five-dimensional beam brightness B5D ¼ 2I=ϵ2ndemanded, but extended the analysis to identify the neededsix-dimensional brightness B6D ¼ 2I=ϵ2nσγ . The recentlypublished design study indicates the upper-bound onallowed energy spread at the injector, which necessitatesa closer look at the physical processes that determine thatenergy spread. In particular, it demands the inclusionof the effects of microscopic space-charge effects asso-ciated with short-range Coulomb interactions such asintrabeam scattering (IBS). Capturing these effects insimulation is, however, extremely difficult as they demanda one-to-one treatment of the beam’s collective effectsat least between neighboring electrons. Such one-to-one

RIVER R. ROBLES et al. PHYS. REV. ACCEL. BEAMS 24, 063401 (2021)

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simulations with a 100 pC beam associated with theapplications analyzed here are computationally unwieldy,and demand a physics-based methodology for capturingthe appropriate effects with a more realistic computationalapproach.In this article we will present a robust design for a

cryogenically cooled rf photoinjector capable of meetingthe ambitious brightness demands of next-generation elec-tron beam applications for both symmetric and asymmetricemittance instruments. Although our primary emphasis willbe on beam dynamics rather than injector engineering, thisbegins with a review of the physics which enables high-field acceleration in normal conducting, cryo-cooled cop-per, as well as the engineering designs of the gun, linac, andsolenoid which allow us to realize these benefits exper-imentally. Once the engineering feasibility is reviewed, wepresent an in-depth analysis of the impact of higherharmonic field content in an injector—in particular inthe context of the present cell design and the high bright-ness of the beams, but also more generally with regard tomodifications to linear beam dynamics and the strength ofdeleterious nonlinear forces. This work is based on earlywork on such effects and also builds on them, lendingunderstanding to their role in the upcoming generation ofhigh-brightness photoinjectors. After this, we will brieflyreview the theory of emittance compensation in standardnonmagnetized injectors so as to develop the language andtools necessary to understand how compensation is modi-fied when the cathode is immersed in a magnetic field.Once we have established the theoretical tools involved

in the design of an injector in the present context, we willpresent multiparticle simulation studies of several relevantinjector working points. The first, which we deem theultrahigh brightness working point, is a design extremelywell suited for driving a short-wavelength compact FEL, asit yields a 19 A beam with 45 nm rad emittance and sub-keV energy spread. This predicted brightness performancenotably exceeds that of even previously studied cryo-cooled guns. We then proceed to a study of this workingpoint scaled down to a lower charge in order to facilitateone-to-one space-charge studies in a computationallyfeasible manner. This scaling, based on the observationthat short-range Coulomb interactions scale with the beamcharge density, allows us to estimate the energy spread frommicroscopic effects in the full 100 pC bunch using a beamwith a charge, and therefore particle number, reduced bythree orders of magnitude. Finally, we introduce an addi-tional bucking solenoid element in the photoinjector designto permit cathode magnetization, and the associated pro-duction of asymmetric emittance beams. We will show thatthe four-dimensional beam brightness ϵ4D ¼ ffiffiffiffiffiffiffiffi

ϵxϵyp can be

preserved down to nearly the level of the ultrahigh bright-ness injector with a subsequent emittance-splitting by aratio of 400. This yields a beam with 4 nm emittance in thesmaller transverse plane at 100 pC bunch charge, which can

be scaled to meet the demands of either a DLA or a linearcollider. The physics involved in such a scaling procedureare also discussed.

II. DESIGN OF PHOTOINJECTOR COMPONENTS

A. Optimized rf structure design

The use of cryogenic copper cavities to reach highelectric fields is motivated first by material properties.At cryogenic temperatures, a number profound changes inmaterial response are noted. First, the power dissipationdue to surface currents is diminished strongly—a factor of4-5 for relevant rf frequencies—by entry into the anoma-lous skin effect regime. This causes the pulsed heatingsuffered by the cavity surface to be ameliorated. Second,the material properties—the coefficient of thermal expan-sion and, to a lesser extent, the thermal conductivity,change in beneficial ways that permit the deposited heatto produce less stress on the rf cavity surface. Finally, theyield strength is greatly increased at cryogenic temper-atures, leading to a greater ability of the structure towithstand the impulse of the electric field and associatedsurface failure. Aspects of this microscopic model havebeen verified, with the role of the magnetic field (pulsedheating) [16] and the electric field [17], respectively,experimentally studied. The definitive study examiningthese effects jointly occurring in an acceleratorlike cryo-genic copper structure is found in Ref. [12].Based on this last study, a proposal was made to employ

this technique in very high field rf photoinjectors, as a wayto increase the brightness. Indeed, the rf gun design weconsider here is a more mature version of that firstintroduced in previous work [13,14], which examined adesign with a field distribution very close to a pure standingwave π-mode with negligible spatial harmonic content. Inthe present work the field profile is modified from this purestanding wave as a result of a detailed optimization of thecell geometry, as is clear from Figure 1. The optimized rfstructure has a very high shunt impedance, owing both tothe increased quality factor Q and to a reentrant geometrythat is feasible in π-mode operation owing to a uniquedistributed coupling architecture [15]. This structure geom-etry is motivated by the goals of minimizing the surfaceelectric and magnetic field strengths. Taken together,achieving these goals results in a structure which cansupport peak on-axis electric fields in excess of500 MV=m without suffering from excessive breakdown(owed to magnetic field-induced heating and electric field-driven stress). Further, we choose a peak design field forinjector operation of 240 MV=m specifically to avoid darkcurrent concerns, which are not a notable issue until thepeak fields reach the threshold of 300 MV=m found inRef. [18]. The residual issues of dark current are planned tobe managed in operations by limiting the rf pulse lengthand employing active sweeping methods. These are

VERSATILE, HIGH BRIGHTNESS, CRYOGENIC … PHYS. REV. ACCEL. BEAMS 24, 063401 (2021)

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evaluated to be adequate solutions, based on the previousexperimental results, for operation at the moderate (100 pCand above) beam charges examined in this paper.On the other side of the trade-off involved in this design,

the field profile is populated by many higher spatialharmonics in addition to the primary resonant wave.This feature of the field profile was not accounted for inprevious iterations of gun design, owing to the lack ofrobust, detailed rf design, and introduces several uniquefeatures to the emittance compensation process which weexplore in this article.Although the present work’s emphasis is in beam

dynamics, we provide a more complete description ofthe rf characteristics in Table I. The point of rf powerrequirements merits some additional clarification. Presentlythe two sections of the gun (full cell and 0.6 cell) are

planned to be fed individually. Each section will beprovided an initial 300 ns fill pulse followed by asubsequent 300 ns pulse for maintaining the fill duringpropagation of the beam. In Table I the values quoted as “A+B” should be understood as corresponding to the powercontained in the first and second of these pulses in each gunsection, respectively. Furthermore, the repetition rate of100 Hz has been chosen primarily for the UC-XFELcontext. At this rate, one dissipates 11 W at cryogenictemperatures, for an estimated cryocooler power of 500 W.This is thus easily feasible, and one can imagine pushing tohigher repetition rates when only the gun operation isconsidered. One final point should be noted about theimpact of rf loading. For the FEL case, there is no need toinject multiple bunches per rf fill, which is the context inwhich loading can become a problem. Multipulse operationis implicit in the linear collider case, however, in which casethe small beam loading effects can be dealt with byadjusting the external rf feed. Additional details aboutthese high gradient structures—their capabilities, manu-facturing techniques, and recent results—can be found inRefs. [15,19] which represent the most current state-of-the-art in published work about these cavities. Further detailsabout the rf studies in progress and planned at UCLA willbe provided in the conclusion.

B. Cryogenic solenoid

To place a solenoid sufficiently close to the rf gun, it isnecessary that it also be located inside the cryostat. It hasbeen decided to employ a normal conducting, cryogeni-cally cooled solenoid to avoid some of the complicationsassociated with superconducting solenoids. However, acrucial consideration in the design of such a cryo-solenoidis to balance the available, temperature-dependent coolingpower versus the resistivity of the windings: a potentiallycatastrophic positive feedback loop where higher localtemperatures lead to greater resistivity and thus powerdissipation, overwhelming the cooling power, must be

FIG. 1. The axial field profile for the full cell design is plottedon top of a cross section of the cell geometry with true-to-lifeaspect ratio.

TABLE I. Several key rf design characteristics are reported.The two numbers quoted for each input power value correspondto the powers of the initial and secondary rf pulses, as explainedin the text.

Parameter Unit Value

Repetition rate Hz 100rf frequency GHz 5.712Operating temperature K 27Input power (FC) MW 10.7þ 3Input power (0.6C) MW 4.8þ 1.6Dissipated energy (FC) J 0.72Dissipated energy (0.6C) J 0.39Shunt impedance MΩ=m 121Pulse length ns 300Quality factor 14000


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avoided [Fig. 2(a)]. To this end, it is necessary to choosea winding material with a low resistivity at cryogenictemperatures; this is usefully summarized by the residualresistivity ratio (RRR) which most often refers to theratio of resistivities between 300 K and 4 K, RRR≡ρ300 K=ρ4 K. Depending on its purity, temper, and otherfactors, copper may exhibit RRR values from 10 to morethan 5,000 [20].

The specific cryo-solenoid design under consideration isshown schematically in Fig. 3. It relies on a conventionaliron yoke, RRR ¼ 2; 000 copper windings, and a coppersarcophagus for mechanical and thermal purposes: thesarcophagus serves to maintain alignment and indexingduring cool down, ensure tight contact between individualwire turns, and, by virtue of being linked to the cryogeniccold head by thermal braids, serve as a heat sink for thewinding Joule heating. The individual windings are to beelectrically insulated with a thin film of polyimide (oftenreferred to as Kapton, a registered trademark of DuPont)which is well characterized at cryogenic temperatures [21].At temperatures of interest, polyimide is approximately fiveorders of magnitude less thermally conductive than copper,therefore, the total thermal resistance between the heatsource (windings) and sink (sarcophagus) is dominated bythese polyimide layers. Electrothermal simulations incor-porating temperature-dependent resistivity are conducted[Fig. 2(b)] to determine the equilibrium thermal distributionand ensure that the runaway scenario described above doesnot occur. Having validated the design’s thermal perfor-mance, field maps were generated using the magnetostaticcode Radia [22]. The final design produces the requisite0.51 T field, employing a current density of 9.2 A=mm2,while dissipating less than 3 watts; operated at roomtemperature, such a solenoid would produce nearly twokilowatts.

III. THE ROLE OF SPATIAL HARMONICSIN RF PHOTOINJECTORS

A. Description of spatial harmonic content in rf fields

It is clear from the square-wavelike field profile that thegun we consider here (again, see Fig. 1) is rich in spatialharmonic content. In the following sections we will discusswhat effects these extra harmonics have on the transverseand longitudinal beam dynamics of emittance compensa-tion. Throughout these sections we will use a Floquetexpansion of the π-mode field to guide the discussion,which defines coefficients an and E0 according to

Ezðz; tÞ ¼ E0ReX∞n¼−∞

an exp½iðnkrfz − ωrftþ ϕÞ�; ð1Þ

and with a−n ¼ a�n,

Ezðz; tÞ ¼ 2E0

X∞n¼1

an cosðnkrfzÞ sinðωrftþ ϕÞ: ð2Þ

By the indicated convention the first harmonic coefficienta1 ¼ 1 (see, for example, [23]). With this convention, E0 isthe accelerating gradient observed by an ultrarelativisticparticle resonant with the first spatial harmonic, havingconstant phase which may therefore yield maximumacceleration. Thus, in the field profile shown in Fig. 1,

FIG. 2. Plots of cryosolenoid thermal performance. (a) Powercurves as a function of temperature for a representative, singlestage Gifford-McMahon cold head and for the cryosolenoid, at auniform temperature. Additional heat loads will reduce theeffective cooling capacity. (b) Equilibrium thermal distributionof winding cross section with 27 K sarcophagus assumed.


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E0 is not actually 120 MV=m (half the peak field) butrather approximately 150 MV=m. The effects of this addi-tional acceleration have not been yet examined in previous,very high field photoinjector studies [4,14].In Fig. 4 we plot the values of the first 13 Floquet

coefficients. We observe that the primary contributions tothe field profile come from the first and third spatialharmonics with a1 ¼ 1 by definition and a3 ≈ −0.2. Aswe will see in the following sections the addition of even asingle strong nonfundamental spatial harmonic is enough tonotably alter both the longitudinal and transverse beamdynamics relative to a pure single harmonic structure.We note in this regard that in recent decades, since the

introduction of high gradient (> 100 MV=m peak field inS-band) rf photocathode guns, the emphasis has been onuse of two-cell structures having negligible higher spatial

harmonic content. This approach has been motivated by adesire to avoid potentially deleterious nonlinear fieldeffects that can be associated with non-speed-of-lightspatial harmonics. These effects are discussed below.Before engaging in this discussion, however, we examinesome positive aspects of structures with higher spatialharmonic content—the introduction of strong second orderfocusing effects and an enhancement of the acceleratinggradient for speed-of-light particles for a fixed peak field.It is interesting to comment that rf structures with

reentrant nose features at the irises were indeed previouslyused in first-generation rf photocathode guns at LANL [1].In these pioneering devices, such rf design features wereemployed to mitigate input power demands; here the samemotivation exists, but it is supplemented by the possibilityof obtaining strong rf-derived focusing in the gun structure,an effect augmented by the foreseen very high fieldoperation. It is also relevant to point out that contributionsto the emittance in the LANL rf photoinjectors due tononlinear fields associated with higher spatial harmonicswere significant in these designs [1]. As wewill show, thesenonlinear effects are greatly reduced when the beam issmall enough—a fact which is naturally realized in anultrahigh brightness injector.

B. Linear beam dynamics with spatial harmonics

The primary transverse effect of higher spatial harmonicsin standing wave accelerating structures is to introducestrong second-order ponderomotive radial focusing forces[24]. The field-normalized strength of these forces ischaracterized by a parameter η defined by

ηðϕÞ ¼X∞n¼1

a2n−1 þ a2nþ1 − 2an−1anþ1 cosðϕÞ; ð3ÞFIG. 4. The Floquet coefficients of the first 13 spatial harmon-ics are plotted for the full cell π-mode structure.

FIG. 3. Render of the cryosolenoid design, showing the iron yoke, copper sarcophagus, and windings.


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where the coefficients an measure the amplitude of theFloquet components of the on-axis axial electric fieldprofile as we previously defined. For pure traveling wavelinacs, η ¼ 0 and for pure single harmonic standing wavelinacs η ¼ 1. For the gun in question,

ηðϕÞ ¼ 1.12 − 0.5 cosðϕÞ: ð4Þ

The nominal value of η for peak acceleration is thus 1.12,i.e., 12% larger than in a pure fundamental standing wavedevice. Furthermore, this strength depends, with a notablylarge coefficient, on the phase-dependent term. In this paperwe will consider primarily on-crest acceleration; however,in the rf gun itself it is not possible to inject the beam on-crest, as the velocity of the electrons changes rapidly withinthe gun. Further refinements of the ponderomotive theorybased on the normalized velocity β < 1, variations in η andrapid relative changes in γ may be useful, but are outsidethe scope of this paper.This effective focusing increase reduces the demands

placed on the focusing solenoid performance. This is awelcome development, as the requirement of using acompact, high field solenoid at relatively short rf wave-length introduces non-trivial challenges in magnet designand implementation, as discussed above.

1. Longitudinal capture dynamics

For high-gradient guns such as the one considered here,the beam is accelerated from the cathode to β near unitywithin the first cell. The longitudinal dynamics in this shortnonrelativistic portion of the gun are well described byutilizing an effective DC field of strength 2μE0 sinðϕ0Þ[25], where μ ¼ P∞

n¼1 an (for our structure μ ≈ 0.8) and ϕ0

is the injection phase. During this period an acceleratingelectron is not perfectly resonant with the fundamentalspatial harmonic. As a result, its phase slips with respect tothe injection phase, and it is also accelerated by interactionwith all present harmonics, giving rise to the direct sumover Floquet coefficients. The phase at which the particlesexit the gun is related to that at which they are injected bythe approximate relationship,

ϕ0 ¼ ϕ −1

2μα sinðϕÞ −1

10μ2α2 sin2ðϕÞ ; ð5Þ

where α ¼ eE0=krfmec2 [26]. As Kim in [26] considers thecase of a pure standing wave, he indicates the maximumelectric field as 2E0. Here, on the other hand, the maximumfield is given by Emax ¼ 2μE0. It is desirable for the beamto leave the gun with the design particle experiencing thephase of maximal acceleration, ϕ ¼ π=2, as this minimizesthe transverse emittance growth associated with the spreadin particle phases [26]. This yields an approximate optimalinjection phase, from the standpoint of minimizing thebrightness degradation from the final iris kick [27],

ϕ0 ¼π

2−

1

2μα−

1

10μ2α2: ð6Þ

It is useful to compare this result to that of the purestanding wave case with μ ¼ 1. Keeping in mind thatμα ¼ Emaxkrf=2mec2, it can be seen that the injection phaseis dependent, in this approximation, on the peak field Emax.This phase is thus not changed in this analysis by thepresence of additional spatial harmonics. As in the analysesof [26,25] which assume that the slippage in phase iscontrolled by the field in the region directly adjacent to thecathode, this situation is indeed expected.The presence of additional spatial harmonics does

change, however, the ratio of the final energy to the peakfield. If we approximate the field in the rf gun as a squarewave instead of a pure cosine form (as motivated by Fig. 1),the total changes in energy due to rf acceleration over thegun length Lg are

ΔU ≃1

2eEmaxLg ðn ¼ 1 onlyÞ; ð7Þ

and

ΔU ≃2

πeEmaxLg ðsquare waveÞ; ð8Þ

respectively. This result could be expected, as the ratio ofthe acceleration field to the peak field deduced from thespatial harmonic analysis, E0=Emax ≃ 0.4 for the squarewave, is nearly identical to the value estimated from Eq. (2).

2. Adjustments of external focusing elements

The standard layout of a modern photoinjector consistsof four basic elements. The first and most fundamental ofthese is the initial high-field rf gun which rapidly accel-erates the beam to a few MeV off of the cathode. This isgenerally followed by a short solenoid which provides aradial focusing kick to guide the beam toward a waistdownstream. The third element is a drift many rf wave-lengths long (meter-scale for the C-band case consideredhere) during which the beam undergoes a full transverseplasma oscillation, arriving finally at the entrance of abooster linac placed such that the beam reaches a space-charge dominated waist near the entrance to the accelerat-ing field.The only mandatory external focusing in this standard

configuration is provided by the solenoid magnet. As wehave already described, higher spatial harmonics in thefield profile leads to strong ponderomotive focusing. Assuch, a photoinjector employing fields with spatial har-monic content may afford to haveweaker solenoid focusingat what is otherwise the same operating point. Alternatively,it can operate with a smaller spot size on the photocathodeas the ponderomotive focusing will mitigate excessive


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growth of the beam size in the gun due to space chargedefocusing. A smaller beam size at injection has the addedbenefit that the intrinsic thermal emittance scales linearlywith the beam size on the cathode [28], such that for allother parameters held constant, the photoinjector may beoperated with lower intrinsic emittance. The ability to focusthe beam directly as it comes off of the cathode is a uniqueadvantage of structure-based rf focusing, as an attempt todo so using solenoidal focusing would magnetize thecathode, which is not acceptable for many commonlyencountered applications.

3. Longitudinal dynamics in downstream linacs

In the present case the structure has been optimized toachieve 240 MV=m peak surface fields on the cathode.This is well below the maximum field achievable due to rfbreakdown (∼500 MV=m) [12], and is limited by choice,in consideration of mitigating dark current and associatedcavity power losses [18]. As a result of the previouslydiscussed factor μ being less than unity, the field associatedwith the primary spatial harmonic is increased relative tothe peak of the field profile. In the downstream linacs onlythis wave remains resonant with the beam and subsequentlyprovides consistent acceleration, so the effective accelerat-ing gradient downstream of the structure is increasedrelative to that of a single harmonic baseline design by25% for μ ¼ 0.8. This is a key advantage of structures withrich spatial harmonic content: a linac structure with peakfield less than its fundamental harmonic contribution canoperate at a substantially larger accelerating gradient than isnominally allowed by surface-field-induced breakdown anddark current considerations. The physical explanation forthis is simply that the walls of the structure see the total fieldat any given moment while an ultrarelativistic beam isdirectly sensitive only to the field of the first harmonic. Thiseffect has a fundamental limit, at least for a π-mode cavity,which is achieved when the field profile is an exact squarewave, a fact which we demonstrate in the Appendix A. Theenhancement factor over the peak physical field for thesquarewave is 4=π, corresponding to an approximately 27%increase in the gradient overwhat onewould expect from thepeak field. The similarity of this ideal figure to the 25%enhancement in the current structure is reflective of thehighly optimized nature of the cell design we consider here.

C. Nonlinear beam dynamics with spatial harmonics

1. Emittance dilution from nonlinear iris kicks

The time-dependent radial momentum kick incurred atthe exit iris of a 1.6 cell gun is awell-known potential sourceof emittance growth. In standard guns with very little spatialharmonic content the kick is perfectly linear in the radialcoordinate, and the emittance growth results only from thephase spread of the electron beam in the rf wave. As wewillshow here, the presence of higher spatial harmonics also

implies the existence of nonlinear terms in this radial kickwhich have the potential to dilute the beambrightness even ifall particles rested at the same rf phase.In this section we compute the radial kick applied to the

beam upon exiting the gun. We do this first for a singlespatial harmonic, and then return to the Floquet expansionto compute the total kick delivered by the full field. Webegin from the Floquet form of the axial electric field in theπ-mode including the radial dependence [23]

Ezðρ; z; tÞ ¼ 2E0

X∞n¼1;odd

an cosðnkzÞ sinðωtþ ϕÞI0ðkn;ρρÞ;

ð9Þwhere the radial wave number is k2n;ρ ¼ n2k2 − ðω=cÞ2 andIn is themodified Bessel function. For now,wewill consideronly one harmonic, n, and leave the summing over spatialharmonics for later. From the longitudinal electric field wecan straightforwardly find the radial electric field and theazimuthal magnetic field using Maxwell’s equations

Eρ;nðρ;z; tÞ¼ 2E0

�annkkn;ρ

�sinðnkzÞsinðωtþϕÞI1ðkn;ρρÞ

Bϕ;nðρ;z; tÞ¼ 2E0

�anωc2kn;ρ

�cosðnkzÞcosðωtþϕÞI1ðkn;ρρÞ:

ð10Þ

The radial force on a speed-of-light particle sampling thesefields is

Fρ;nðρ;z; tÞ¼−eðEr;n−cBϕ;nÞ

¼−2eE0ankkn;ρ

I1ðkn;ρρÞ½nsinðnkzÞsinðωtþϕÞ

− cosðnkzÞcosðωtþϕÞ�: ð11ÞKeeping with the traditions of similar calculations [29],

we extract the radial kick as a function of location in thegun in the impulse approximation, with ρ held constant,assuming that β ¼ 1 after the first 0.1 cell and taking z ¼ 0to coincide with the end of that first 0.1 cell region in thecase of a 1.6 cell gun. Despite our explicit interest in the 1.6cell case, this analysis holds in general for a gun of anynumber of cells. By integrating the radial force expressionand using the fact that t ≈ z=c, we obtain

Δpρ;nðρ;zÞ¼Z

z

0

dz0

cFρ;nðρ;z0; tÞ

¼−2eE0anI1ðkn;ρρÞ

ckn;ρðsinðϕÞ

−fsin½ð1−nÞkzþϕ�þ sin½ð1þnÞkzþϕ�gÞ:ð12Þ


063401-8

For the kick at the end of the 1.6 cells of the gun weevaluate this expression at z ¼ 3λ=4 ¼ 3π=2k. This gives

Δpρ;nðρÞ ¼ −2eE0 sinðϕÞan

kn;ρcI1ðkn;ρρÞ; ð13Þ

where we have employed the fact that we are interested inonly odd n. With this single harmonic kick we can evaluatethe full kick via a sum over each contributing spatialharmonic,

ΔpρðρÞ ¼ −ec2E0 sinðϕÞ

Xn

ankn;ρ

I1ðkn;ρρÞ: ð14Þ

We therefore find the kick in the angular radial coordinate,

Δρ0 ¼ Δpρ

pz¼ −

2eE0

γmc2sinðϕÞ

Xn

ankn;ρ

I1ðkn;ρρÞ: ð15Þ

where γ here is in particular the beam energy at the gun exit.To extract the lowest-order correction to the common linearapproximation of the radial dependence, we expand to themodified Bessel function describing this dependence, usingI1ðxÞ ≈ ðx=2Þð1þ x2=8Þ, to arrive at

Δρ0 ≈ −2eE0 sinðϕÞρ

2γmc2Xn

an

�1þ k2n;ρρ2

8

�ð16Þ

¼ −2eE0 sinðϕÞρ

2γmc2Xn

an

�1þ

�n2k2 −

ω2

c2

�ρ2

8

�ð17Þ

¼−eEzð0ÞsinðϕÞρ

2γmc2

�1−

ρ2ω2

8c2

�1þ c2

ω2

E00z ð0Þ

Ezð0Þ��

: ð18Þ

We retrieve the familiar linear kick in the first term [29],while the second term represents a third-order kick whichvanishes if there are no nonresonant spatial harmonicspresent. This term is suppressed by a factor ρ2ω2=8c2

relative to the linear term. For reference, at 1 mm off-axis ina C-band structure, this suppression is approximately threeorders of magnitude.This radial angular kick corresponds to identical x and y

angular kicks,

Δx0 ¼ −eEzð0Þ sinðϕÞx

2γmc2

�1 −

ρ2ω2

8c2

�1þ c2

ω2

E00z ð0Þ

Ezð0Þ��

:

ð19Þ

We use this expression to estimate the emittance growththat results from the nonlinear term. For simplicity weassume the beam to be at a waist at the exit of the gun. Forconciseness of notation we will write the kick in the formΔx0 ¼ axð1 − ρ2δÞ, the relevant beam moments become

hx02i ¼ σ2x00 þ a2σ2x0 − 8a2δσ4x0 þ 21a2δ2σ6x0 ð20Þ

hxx0i ¼ aσ2x0 − 4aδσ4x0: ð21Þ

From this result we easily find the emittance

ϵ2x ¼ σ2xσ2x0 − σ2xx0

¼ ϵ2x0 þ a2σ4x0 − 8a2δσ6x0 þ 21a2δ2σ8x0 − a2σ4x0

− 16a2δ2σ8x0 þ 8a2δσ6x0: ð22Þ

In this expression we see the expected cancellation of thelinear order terms, as such linear effects do not causeemittance growth, and interestingly we also see cancella-tion of the terms coupling the linear and nonlinear kicks.All that remains are the purely nonlinear contributions:

ϵx ¼ ϵx0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 5a2δ2σ8x0

ϵ2x0

s: ð23Þ

It is insightful to write down these terms in non-normalizedvariables

aδσ4x0ϵx0

¼�eEzð0ÞsinðϕÞ

mc2

�βx0ðkσx0Þ2

16γ

�1þ c2

ω2

E00z ð0Þ

Ezð0Þ�: ð24Þ

The first quantity in Eq. (24) is the accelerating gradient inunits of electron rest energy; the second term contains thebeam beta-function at the gun exit and also a termcomparing the scale of the rf wavelength to the transversebeam size. The last quantity indicates the relevant nonlineareffects of spatial harmonics,

1þ c2

ω2

E00zð0Þ

Ezð0Þ¼ 1 −

Pnn

2anPnan

: ð25Þ

Let us evaluate this effect for the current case. The sum overharmonic contributions is roughly 1.35. As a numericalexample, we will take roughly the values corresponding tothe injector design presented in Sec. V: with σx ¼ 325 μm,γ ¼ 14, ϵnx0 ¼ 45 nm, and the gradient 240 MV=m, wefind a roughly 1% increase in the emittance at 45 nmcorresponding to a sub-nm growth in the normalizedemittance. This is nearly ignorable value, largely as aresult of the relatively small beam size at the exit of the gun.This result is quite encouraging for the case we consider

in the FEL injector section, where the beam parameters areas indicated above. However, we would eventually like toconsider higher charge cases, in which the implicit scalingof the beam dimensions as Q1=3 implies that the emittancegrowth term in Eq. (24), assuming ϵx0 ∝ σx, grows as Q2.Indeed, for two cases of interest—linear collider (discussedbelow) and also wakefield accelerator drivers—we shouldeventually consider increasing the charge by an order ofmagnitude. In such cases the nonlinear contribution to the


063401-9

emittance from higher spatial harmonic focal effectscontributes nearly equally to the final emittance as com-pared to more familiar effects (due to space-charge,thermal, chromatic considerations). For the moment, how-ever, we concentrate on lower charge, higher brightnessexamples, which are relevant to the operation of the newC-band high gradient photoinjector initiative at UCLA thataims to enable new possibilities in both FEL and linearcollider applications.

IV. THE RMS ENVELOPE EQUATION FORACCELERATING BEAMS

In order to prepare for the discussions of simulationresults that follow, we introduce here the rms envelopeequation for accelerating beams, and examine the behaviorof the solutions in various relevant limits. This discussionserves the purpose of elucidating the results we will presentlater which are found via numerical simulations. Theestablished approach to analyzing the dynamics of emit-tance compensation [2] utilizes longitudinal slices of thebeam, designated by the variable ζ ¼ z − vbt ≃ z − ct. Thisanalysis assumes that each axially symmetric slice evolvesnearly independently under the rms envelope equation,which in the limit vb ≃ c has the form

σ00xðz; ζÞ þγ0

γσ0xðz; ζÞ þ

�ηþ 2b2

8

��γ0

γ

�2

σxðz; ζÞ

¼ ϵ2nγ2σxðz; ζÞ3

þ IðζÞ2I0γ3σxðz; ζÞ

: ð26Þ

Here we indicate the derivative with respect to the distancealong the beam propagation direction with a prime symbol,d=dz ¼ ðÞ0, and the constant rate of change of γ due tointeraction with the resonant (speed-of-light) spatial har-monic is γ0 ¼ eE0=mec2. Also, the local current IðζÞ isnormalized to the Alfven current I0 ¼ ec=re, the (constantunder linear transformations) rms normalized emittanceϵn ¼ γ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihx2ihx02i − hxx0i2

pand b ¼ cBz=E0. Finally, we

note that the rms envelope equation self-interactionterm explicitly includes only linear forces, with nonlineareffects implicitly entering only through the slice-emittanceevolution.This emittance term on the right-hand side of Eq. (26)

derives mainly, however, from the thermal-like rms spreadin transverse momenta in the beam. Further, it is traditionalto follow the beam dynamics governing the emittancecompensation process by ignoring this term, which forrelevant parameters of high brightness beams (large I, smallϵn) from rf photoinjectors is small compared to the space-charge derived term. In the present work, we will examineanother scenario, in which the beam is magnetized uponemission, by placing a nonvanishing solenoidal field on thephotocathode, Bc. In this case, one transforms the rmsenvelope equation to

σ00xðz; ζÞ þγ0

γσ0xðz; ζÞ þ

�ηþ 2b2

8

��γ0

γ

�2

σxðz; ζÞ

¼ ϵ2n þ L2

γ2σxðz; ζÞ3þ Iðz; ζÞ2I0γ3σxðz; ζÞ

; ð27Þ

where L ¼ eBcσ2x;0=mec is the canonical rms angular

momentum owed to the cathode magnetic field. Uponleaving the solenoid region near the rf gun, this canonicalangular momentum is converted into mechanical angularmomentum. We will be concerned with cases whereL2 ≫ ϵ2n, and the term proportional to σ−3x may no longerbe ignored in the envelope evolution.From this equation it would appear that the canonical

angular momentum behaves identically to an increasedthermal emittance. This is correct from the viewpoint of theenvelope behavior, but hides important differences in themicroscopic dynamics. In an emittance-dominated beam,the dynamics are thermal, and phase-space trajectoriescross (they are nonlaminar). In the case of an angularmomentum-dominated beam, however, the particle flow islaminar, and undergoes rotations. This laminarity is ashared trait of space-charge-dominated beams. As a result,the angular momentum differs from the thermal emittancein that it is physically realized in linear correlationsbetween the two transverse planes, which may in principlebe removed by suitable downstream beamline elements.In the following sections, we examine particular solu-

tions of the envelope equation in the space-charge-domi-nated and angular momentum-dominated limits.

A. Space-charge-dominated beam behavior withacceleration

In the limit that the beam is space-charge dominated andpossesses no appreciable angular momentum, and there isno local solenoid field (b ¼ 0), we omit the angularmomentum and emittance terms in Eq. (26), and theresultant differential equation admits a particular solution,for current I, known as the invariant envelope [2]

σx;invðzÞ ¼2

γ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI

I0ð2þ ηÞγ

s: ð28Þ

The invariant aspect of this behavior is the phase spaceangle of the envelope [2],

Θσ ≃γσ0x;invσx;inv

¼ −γ0

2: ð29Þ

It should be noted that this corresponds exactly to the firstangular kick due to entry into the linac fields [24],

Δx0 ¼ −γ0x2γi

; ð30Þ


063401-10

where γi is the initial value of γ in the linac. Thus thematching of the beam to the invariant envelope in the postacceleration linac is accomplished by injecting the correctinitial value [γ ¼ γi in Eq. (28)] at a waist (σ0x ¼ 0).The emittance compensation process begins before

injection into the linac, with the beam focused by a solenoidto the waist described above at the linac entrance that issimilar in transverse size to that found at emission from thephotocathode. At this point, the errors in trace spaceorientation between the various ζ-slices are minimized,leaving an emittance that is a local minimum [30]. It is,however, not a global minimum, as the addition of the post-acceleration linac further decreases the emittance, whileadding energy to the point of the “freezing” of the space-charge induced effects, which scale in strength as γ−2.The diminishing of the emittance due to compensation inthe linac is due to two effects: the further rearrangementof the slices’ relative trace-space orientation; and the slowreduction of the beam size as it tends to follow the invariantenvelope. In order to understand the behavior of thenonmatched slices, one should examine the perturbedenvelope equation for conditions close to the invariantenvelope.This analysis begins with the writing of the perturbed

envelope equation [2], wherewe consider envelope behaviornear the invariant solution, σxðz;ζÞ¼ σx;invðz;ζÞþδσxðz;ζÞ.This can be written as

δσ00xðz; ζÞ þγ0

γδσ0xðz; ζÞ þ

�1þ η

4

�γ0

γ

�2�δσxðz; ζÞ ¼ 0:

ð31Þ

We note that this differential equation is now independent ofcurrent I. The solution for a slice beam envelope having anerror in rms size δσx;0ðζÞ is

δσxðz; ζÞ ¼ δσx;0ðζÞ cos� ffiffiffiffiffiffiffiffiffiffiffi

1þ ηp

2ln

�γ

γi

��; ð32Þ

with derivative

δσ0xðz;ζÞ¼−ffiffiffiffiffiffiffiffiffiffi1þη

p2

γ0γiγ

δσx;0ðζÞsin�1þη

2ln

�γ

γi

��: ð33Þ

FromEqs. (32) and (33), we can deduce that the area in tracespace scales as γ−1, and the phase space area is conserved.This area we term the offset emittance, and designate it asϵδ ≈ γjhδσx;0iζhδσ0x;0iζj, where h:iζ indicates an average overbunch slicesweighted by the current distribution. For awell-optimized design, this should be close to the thermalemittance at the cathode, perhaps increased by effectsassociated with rf and nonlinear space-charge.The normalized beam emittance consists of a contribu-

tion from this offset emittance, magnified by the distance

from the phase space origin to its center—the invariantenvelope. We may write an approximate expression for thebehavior of the normalized emittance as

ϵn ≃ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵ2δ þ ðγσx;invσ0x;invÞ2

q

≃

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵ2δ þ

4I2

ð2þ ηÞ2I20γ02γ2s

: ð34Þ

This expression ignores the dynamics of slice realignmentwith respect to the invariant envelope trace space direction,but captures the overall behavior—the normalized emit-tance approaches an asymptotic value ϵδ due to the seculardiminishing of the beam envelope.

B. Angular-momentum-dominated beambehavior with acceleration

In an accelerating beam, because of the relatively strongdecrease in the space-charge forces as a function ofincreasing energy, a beam eventually becomes emittancedominated as it attains high enough energy. In this case, onemay ignore the term proportional to the peak current inEq. (26), and write a particular solution of the resultingequation as

σϵ;x ¼�8

η

�1=4

ffiffiffiffiffiϵnγ0

r: ð35Þ

Because of opposing γ-dependencies of the focusing andadiabatic damping of the emittance, the emittance-domi-nated solution yields simply a constant beam size. Thisstands in contrast to the secularly diminishing invariantenvelope solution for space-charge-dominated beams. Notealso that formally we may apply this result to the angular-momentum dominated case by substituting L for ϵnin Eq. (35).In the space-charge-dominated case, one chooses to

operate at the particular solution of the envelope equationfor two reasons: it is associated with a phase space anglewhich is independent of the local current, and it ismonotonically decreasing at a rate sufficient for diminish-ing of the correlated emittance. This is clearly not the casein the angular-momentum-dominated case. The constantbeam size associated with the particular solution in this casewould not facilitate the emittance approaching the offsetemittance asymptotically, rather once the beam becomescompletely angular-momentum-dominated the space-charge oscillations no longer play a role in the envelopedynamics. This is highly undesirable, and as such oneshould inject the beam into the linac at a size large enoughthat the beam is still partially space-charge dominated, andin particular at a size which is notably larger than theparticular solution indicated by Eq. (35). This naturallyprevents one from operating at the optimal spot size usually


063401-11

reached for a space-charge-dominated beam which isideally close to the spot size at the cathode. This is onefundamental reason why emittance compensation withangular momentum-dominated beams is naturally lessefficient. In fact, in practical scenarios, the beam spendsmuch of its time in the injector partially space-charge-dominated, but in such a way that angular momentumcannot be ignored.There are yet other physics concepts at play which

distinguish emittance compensation with angular momen-tum from that without. In fact, there are two sources whichcontribute to the inefficiency of emittance compensation inmagnetized photoinjectors. The first is that the rmsenvelope equation with angular momentum has no solutionwhich has a phase space angle independent of the localcurrent for all energies, as we show in Appendix B. There isthus no clear working point analogous to the invariantenvelope which achieves ideal compensation throughanalytical methods, rather the beam envelope must beoptimized numerically. To understand the second sourceof inefficiency we must consider, instead of mismatch inthe phase space angle, mismatch in the local beam size. Inthe standard, non-magnetized case, dependence of the localslice beam size on the current contributes to what weidentified already as the offset emittance, and part of thegoal of numerical optimizations is to minimize this mis-match at the booster entrance. In the case of a magnetizedphotoinjector, reduction in the beam size mismatch isinherently less efficient. This is because in addition tothe usual contributions to the emittance, a mismatch in thebeam size also implies that there is no rotating frame inwhich every slice ceases to rotate [31]. This results from thefact that all slices are born with the same size and thus thesame canonical angular momentum, and will thereforerotate at different rates if they have different sizes furtherdownstream. Since in practice one can never achieveperfect matching of all slice sizes, the process of minimiz-ing the offset emittance is inherently less efficient.In the following sections we will give some quantitative

analysis of the beam behavior in various parts of theinjector and in particular discuss the ramifications of thissection’s discussion on the ideal injector operating point inthe presence of angular momentum.

C. Beam dynamics with angular momentumin the prebooster drift

After leaving the compensating solenoid the beam ismoderately space-charge dominated, in order that theemittance oscillations may proceed toward their eventualsecond minimum. In particular, for reasons elucidated inthe previous section, one should ensure that the beamremains space-charge dominated until it enters the boosterlinac downstream. Since the angular momentum here is, byassumption, large, overfocusing could cause the beam totemporarily become angular-momentum dominated near

the transverse waist, thereby interfering with the evolutionof the emittance oscillations. In this section we will studythe dependence of the size of the beam waist on the beamparameters upon exiting the solenoid. In this region thebeam satisfies the envelope equation in the absence ofacceleration and focusing,

σ00 −L2

γ2σ3−

I2I0γ3σ

¼ 0: ð36Þ

This regime is particularly difficult to deal with analyticallyfor two reasons. First of all, as is expected, this equationdoes not have an analytic solution in general which is not anunwieldy infinite series. Second, although the beam ismoderately space-charge dominated upon exiting thesolenoid, the two terms can contribute nearly equally nearthe beam waist even in an optimized design, as we will seebelow. Nevertheless, we may determine the size of thebeam at the waist by extracting a conservation law from theequation of motion. We accomplish this by multiplyingthe envelope equation by σ0 and interpreting the subsequentterms as exact derivatives,

ddz

�1

2ðσ0Þ2 þ L2

2γ2σ2−

I2I0γ3

logðσÞ�¼ 0: ð37Þ

This implies that the quantity in brackets is conserved alongthis section of the transport. In particular then, we mayrelate the waist beam size σmin to the parameters at thesolenoid exit σ0 and σ00 as

σ020 þL2

γ2σ20−

2I2I0γ3

logðσ0Þ¼L2

γ2σ2min

−2I

2I0γ3logðσminÞ: ð38Þ

Rewriting this in a slightly more interpretable form,

2I0γ3σ020I

¼2I0γL2

Iσ20

σ20σ2min

�1−

�σmin

σ0

�2�−2log

�σmin

σ0

�: ð39Þ

In this form we may identify the term S≡ 2I0γL2=Iσ20 asthe relative strength of the space-charge and angularmomentum terms in the envelope equation at the exit ofthe solenoid. If we additionally define A≡ 2I0γ3σ020 =I andx ¼ σmin=σ0, this relation can be written in simpler form as

A ¼ Sx2

ð1 − x2Þ − 2 logðxÞ: ð40Þ

The solution x to this equation provides the size of the beamat its waist relative to the size at the exit of the solenoid. Ofparticular interest is the case when the beam waist corre-sponds to the beam size at which the angular-momentumand space-charge terms become equivalent, as past thispoint one can consider the angular-momentum term dom-inant, thereby interfering with the emittance compensation


063401-12

process. Under this condition, x0 ¼ ðL=σ0Þffiffiffiffiffiffiffiffiffiffiffiffiffi2I0γ=I

p ¼ffiffiffiS

p. Thus A ¼ 1 − S − 2 logð ffiffiffi

Sp Þ. If A exceeds this value,

the beam over-focuses and becomes angular momentumdominated at the waist. As a result, one should design theinjector such that A is below this cutoff. In the simulationsstudies we present below, we find through numericaloptimization that in fact the ideal scenario is to operatesuch that xmin ≈

ffiffiffiS

p. This is unsurprising: in general the

emittance compensation process is more efficient withsmaller beam size and this represents the smallest beamsize that allows emittance compensation to proceed withoutinterruption.

D. Beam behavior in the booster linacof a magnetized photoinjector

In a traditional high-brightness (unmagnetized) injectorthe aim of the booster linac is to facilitate the transition ofthe beam from space-charge dominance to emittancedominance without interrupting the final stage of emittancecompensation, in particular by optimally damping thetransverse beam size with increasing energy. This increasein energy serves to diminish the relative strength of thespace-charge forces. To optimally make the transition toemittance domination, one operates at or near the invariantenvelope, where the beam size drops as σx ∝ 1=

ffiffiffiγ

p, and the

dominant slice dynamics are independent of the local beamcurrent, thereby achieving the goal of allowing the emit-tance compensation dynamics to proceed to their eventualminimum-emittance configuration. As we demonstrated inprevious sections, such a solution does not exist in thepresence of notable emittance or angular momentumeffects. Furthermore, the predominance of angular momen-tum in the magnetized beam case implies that the describedtransition occurs at a much lower energy, thereby occurringearlier for equal accelerating gradients. In order to operateat moderate gradients while arresting the emittance evolu-tion near the minimum of the emittance oscillations, thisimplies placing the booster linac slightly further down-stream than the transverse beam waist—consistent with thequalitative analysis of [31]. Furthermore, to facilitate thediminishing of the emittance we would like to employ analternative approach to reducing the beam size, inspired bythe 1=

ffiffiffiγ

pdependence of the invariant envelope. This

indeed implies operation far from the equilibrium beamsize associated with the pure angular-momentum-domi-nated solution, thus an accurate analytic description of theenvelope progression for ideal compensation in this casedemands a relatively elaborate approximation scheme.Here we present a simple model for the beam envelope

evolution in the relevant regime, where the beam enters thebooster under space-charge dominated conditions, butbecomes angular momentum dominated rapidly thereafter.This analysis is intended to elucidate the dominant proc-esses involved in the envelope evolution in the booster linac

of an injector with angular momentum. It will also serve toprovide a method for describing how the beam transitionsfrom a space-charge-dominated state to an angular-momen-tum-dominated one.At entrance into the booster linac the space-charge

contribution to the envelope equation only slightly exceedsthat from angular momentum. This situation is describedclearly by Fig. 5 where we have plotted the ratio of thespace-charge and angular momentum terms in the envelopeequation through the injector design presented in moredetail in Sec. VII. As discussed above, at the beam waist thespace-charge and angular momentum terms contributenearly equally to the envelope oscillations, and the optimalsolution requires allowing the beam to expand to a state inwhich it is more heavily space-charge dominated beforeinjecting it into the linac. When the beam does enter thebooster at roughly z ¼ 1.6 m, the transition to angular-momentum-dominance is rapid due to the effects ofacceleration and occurs before the beam size changesappreciably. As such, we anticipate that the contributionof space-charge to the beam dynamics may be sufficientlycaptured by approximating the space-charge term in theenvelope equation by evaluating it only employing thebeam size at linac entrance,

σ00 þ γ0

γσ0 þ η

8

�γ0

γ

�2

σ ¼ L2

γ2σ3þ κ

γ3σ0: ð41Þ

Since we assume the dynamics along the majority of thelength of the linac will be dominated by the angularmomentum contribution, we will separate the envelopesolution into two terms: σðzÞ ¼ σAMðzÞ þ σSCðzÞ. We willconsider space-charge as a perturbation such that σAMðzÞsatisfies

FIG. 5. The ratio of the space-charge and angular momentumterms in the envelope equation is plotted along the injectorstudied in Sec. VII.


063401-13

σ00AM þ γ0

γσ0AM þ η

8

�γ0

γ

�2

σAM ¼ L2

γ2σ3AMð42Þ

This equation has an exact solution, given by [32]

σ2AMðzÞ ¼ σ2i cos2ðψðzÞÞ þ 4

ffiffiffi2

pσiγiσ

0i

γ0ffiffiffiη

p sinðψðzÞÞ cosðψðzÞÞ

þ 8

γ02η

�L2

σ2iþ γ2i σ

0i2

�sin2ðψðzÞÞ; ð43Þ

where

ψðzÞ ¼ffiffiffiη

8

rlog

�γ

γi

�: ð44Þ

Here, in the absence of space-charge forces, σi and σ0iwould be the transverse size and angle at the start of thelinac, but for now they should be thought of as arbitraryvalues which we will fix after describing the space-chargecontribution.Next, we show that there is a more accurate way to

approximate the dynamics, by modifying these initialconditions in the angular momentum envelope beyondinclusion of the space-charge term in the constant beamsize approximation. Returning to this space-charge term,we determine the evolution of σSC by linearly expandingEq. (41) about σAMðzÞwhile neglecting the residual angularmomentum term, a choice which we justify retroactively bydemonstrating the validity of these approximations throughnumerical integration of the envelope equation. With thischoice, we now write

σ00SC þ γ0

γσ0SC þ η

8

�γ0

γ

�2

σSC ¼ I2I0γ3σ0

: ð45Þ

The particular solution to this equation is

σSCðzÞ ¼4I

I0γ02γð8þ ηÞσ0ð46Þ

We further note that this does not have σSCð0Þ ¼ 0 andσ0SCð0Þ ¼ 0, thus if we assume σi ¼ σ0 and σ0i ¼ σ00 inEq. (43) we would be contradicting our initial conditions.We can resolve this in one of two ways: either including thehomogeneous solution in the space-charge solution andchoosing its coefficients accordingly, or by taking, inEq. (43),

σi ¼ σ0 −4I

I0γ02γið8þ ηÞσ0σ0i ¼ σ00 þ

4II0γ0γ2i ð8þ ηÞσ0

: ð47Þ

With this choice our initial conditions are consistent for thetotal beam envelope σðzÞ ¼ σAMðzÞ þ σSCðzÞ.

The seemingly arbitrary choices made in developing thisderivation demand some justification, which we provide inFig. 6. In this figure the beam and accelerator parametersused correspond to those for the first booster linacpresented in Sec. VII. In it we have plotted four differentsolutions to the envelope equation. The first, the blue linewhich is concealed by the red line, is obtained by numeri-cally integrating the envelope equation with no approx-imations or assumptions. The second, labeled “Analytic,No SC” is simply Eq. (43) with σi ¼ σ0 and σ0i ¼ σ00. Thethird, labeled “Analytic, Hom. SC” is the sum of theanalytic angular momentum and space-charge envelopeswith σi ¼ σ0 and σ0i ¼ σ00 and the homogeneous termsincluded in Eq. (46) to make σSCð0Þ ¼ 0 and σ0SCð0Þ ¼ 0.The final line, labeled “Analytic, Inhom. SC” is the solutionwe have presented as optimal in the derivation. This claimappears to be fully justified by the figure, where thissolution lies on top of the numerical integration. Theanalytic solution ignoring space-charge finds a transversewaist which is both too small and occurs too early due tothe complete absence of the space-charge defocusing force.The solution with homogeneous terms maintained in thespace-charge contribution approaches a waist too early andwith too large of a minimum value.Unlike the result of the previous section, this analysis does

not rely on the beam entering the linac near the constantenvelope solution associated with the angular-momentum-dominated dynamics. For the reasons stated above, this ismuch more relevant to the practical optimization of aninjector with a high degree of angular momentum.

V. ULTRA-HIGH BRIGHTNESS MODE FOR ANXFEL DRIVER

A. Beam requirements for an advanced XFEL

We next concentrate on a highly relevant example ofhigh brightness, unmagnetized beam production in the high

FIG. 6. The beam envelope in the booster linac presented inSec. VII is determined four different ways, with each described inthe text.


063401-14

field photoinjector, which is applied to a frontier x-ray freeelectron laser (XFEL) scenario. In this regard, recent worktoward the design and realization of an ultracompact XFEL(UC-XFEL) with a footprint on the order of 40 m hashighlighted the necessity of producing beams with excep-tional 6D brightness [4]. The existence of such a device islargely predicated on the ability to use ultrahigh gradientaccelerating structures to produce a GeV-scale beam in only10–20 meters. The linac design required to achieve thisgoal has largely been demonstrated [15], and further studiesaimed at realizing mature devices based on this technologyare currently being pursued by a SLAC-UCLA-LANL-INFN collaboration [4]. Nevertheless, scaling the energy ofthe electron beam down to 1 GeV while fixing the x-rayphoton energy demands a notable, ambitious decrease inthe electron beam normalized emittance to enable robustlasing. Thus, the state-of-the-art status of free-electron laserphotoinjectors, producing some 10’s of ampere beams with0.2 μm rad scale normalized emittances, is not adequate torealize an ultracompact XFEL on the desired scale.In previous work [13,14] it was determined via simu-

lations employing ideal rf and solenoid field maps that witha 240 MV=m peak field on a photocathode, an electronbeam of 55 nm rad normalized emittance and 17 A currentcould be produced. It has additionally been demonstrated insimulation that this exceptional brightness can be preservedwith minimal degradation during transport, acceleration (to1 GeV) and longitudinal compression [4,33] by a factor of200. The ultrahigh brightness operating regime for thepresently considered photoinjector serves to explore andvalidate the ability to produce beams of similar quality tothose proposed in previous iterations of the gun when amore complete view of the rf and solenoid fields used isemployed. Further, we deepen the previous studies byincluding a simulation study of an important effect in suchunprecedented 6D brightness systems, that of intrabeamscattering (IBS) [34]. This effect, which until now has notbeen examined by detailed simulations of the beam’smicroscopic behavior, may have notable negative implica-tions for 6D phase space dilution—introduction ofunwanted additional slice energy spread.It is also worth noting that constraints similar to those of

the UC-XFEL regarding beam brightness can be found inlarge-scale XFEL applications demanding high photonflux at high photon energy, such as the proposed MaRIEXFEL. A design study of this particular potential machinefound that a beam source of the quality we seek todemonstrate here would substantially decrease the diffi-culties associated with reaching these extreme perfor-mance metrics [5].

B. Comment on thermal emittance

In order to understand the low emittance presented in thesimulations to follow, a discussion of the factors whichenable such a low emittance is demanded. This discussion

naturally begins with the thermal emittance at the cathode.We recall that this has the form [28]

ϵn;th ¼ σx

ffiffiffiffiffiffiffiffiffiffiffiMTE3mc2

rð48Þ

where the mean transverse energy MTE is defined byMTE ¼ ℏω − ϕeff where ℏω is the photon energy of thecathode drive laser and ϕeff is the effective cathode workfunction after the Schottky effect has been accountedfor [35]. In the present scenario we envision an MTE of140meV, consistentwith a cathode drive laser of 262.1 nm inthis simplemodel. This results in a thermal emittance, quotedin the usual way, of 0.3 μm=mm. This value, althoughslightly ambitious, is not far from the state-of-the-art forcopper cathodes in photoinjectors [36], and iswell above thatachieved at test stands [37]. Further, it has been proposed thatnotable improvements may be made with cryogenic oper-ation of the injector and thus the photocathode [38]. This andother approaches promise to lower the thermal emittance inthe near future, and the subject of exploiting much smallerMTE is actively being explored at present [39].The use of low thermal emittances to enable the high

brightness we seek here implies a small beam size on thecathode of roughly 100 μm, assuming adequate quantumefficiency. This question of the impact of and questionssurrounding quantum efficiency will be discussed withinthe experimental context in the conclusion. This small spotsize injection is enabled by several features of the injectordesign presented already: namely strong ponderomotive rffocusing and rapid acceleration in a high-field environment.Furthermore, since the emittance is already so small, anyunexpected increases to the thermal emittance wouldroughly add in squares to the optimal value achieved insimulations below and would be unlikely to require muchin the way of changes to the injector operating point weconsider. This is expected to be the case as long asemittance does not grow large enough so as to jeopardizethe space-charge-dominated nature of the envelope evolu-tion. Thus even with more standard thermal emittancefigures we would not expect the final emittance to exceed55 nm rad, which is the value found in previous iterations ofthis very gun design. The reason for this is that the finalemittance is determined roughly in equal parts by thermalemittance considerations and other factors such as rf andnonlinear space-charge emittance growth.

C. Photoinjector performance

The simulations for both the high-brightness operatingconditions and those relevant to flat beam (magnetizedphotocathode) operation are performed using the GeneralParticle Tracer (GPT) code with 350k macroparticles withthe “accuracy” parameter set to six [40]. In these initialstudies we employ a mesh-based three-dimensional space-charge routine suitable for capturing the complex collectivephysics in the gun but which elide over microscopic


063401-15

space-charge effects such as IBS and disorder-inducedheating (DIH) [41]. These effects are discussed in detailin later sections. For both operating modes, the beamdistribution is uniform in the longitudinal coordinate and aGaussian cutoff at 1σr in the transverse coordinates. Thehigh-brightness operating mode of the injector consists ofthe high-gradient 1.6 cell gun, whose complete on-axisfield profile in shown in Fig. 7, a cryogenically cooledcompensating solenoid, and a 40 cell C-band booster linacstructure of roughly one meter length [15].The ultrahigh brightness working point has been opti-

mized to generate the smallest possible normalized emit-tance with just under 20 A peak current at 100 pC bunchcharge, as was found in previous, less mature iterations ofthe gun design. The final parameter set, found throughnumerical minimization of the beam emittance, is given inTable II. The result of this optimization is found in Fig. 8.

Here we show the normalized emittance in blue alongsidethe root-mean-square spot size in red as the beam travelsthrough the injector. In addition, we include graphicsdisplaying the locations of the gun, the solenoid, and the1 m-long booster linac. We note that in these simulations,we have employed the spatial harmonic-rich structuredescribed in detail above for both the gun and booster linac.The emittance compensation profile evolution is rela-

tively standard, with injection into the linac occurring at theusual “Ferrario working point” at which the beam enters thebooster at its waist. Further, the beam is very nearly onthe invariant envelope corresponding to the parameters ofthis high gradient standing wave linac. This invariantenvelope initiates at a 64 μm spot size waist accordingto Eq. (28), which is nearly exactly the 63 μm waist foundthrough numerical optimization. As a result we observe theexpected simultaneous damping of both the emittanceand the beam size, as described qualitatively above—the emittance has a value of roughly 45 nm rad as it exitsthe booster linac, a 10 nm rad improvement over previousiterations of the gun. This enhanced performance may beattributed to the beneficial effects associated with spatialharmonic content in the gun and linac. Furthermore,the transition to emittance-dominated beam transport isalmost ideally accomplished in this first booster linac, asthe beam finds an asymptotic value of 37 μm, whichis almost exactly equal to the quadrature sum of theinvariant envelope and the emittance dominated solu-

tion σϵ;x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið8=ηÞp

ϵn=γ0q

.

We also highlight the dramatic expansion of the trans-verse distribution through the gun and solenoid, at the peakof which the beam has grown relative to its size at thecathode by roughly an order of magnitude. This is con-sistent with the findings of the authors of Ref. [30], whoshowed that emittance compensation is most effective when

FIG. 7. The on-axis field profile of the 1.6 cell gun design isshown.

TABLE II. The injector parameters relevant to emittancecompensation for ultrahigh brightness operation are listed.


Charge pC 100Laser spot size (Pre-Cut) μm 151Laser spot size (Post-Cut) μm 76Injection phase ° 44Laser length ps 5.8Peak cathode field MV=m 240Solenoid field T 0.51Solenoid FWHM cm 7.4Solenoid center cm 12.5Booster gradient MV=m 77Booster entrance m 1.165Booster phase ° 90

FIG. 8. The emittance and beam size evolution of the ultrahighbrightness operating point are shown through the firstbooster linac.


063401-16

the beam propagates far from equilibrium. In this way, localphase space wave-breaking due to nonlinear field effects isavoided. This rapid expansion also serves to rearrange thebeam distribution to produce more linear self-fields,through the transverse “blowout” effect.We next explore the beam distribution at the linac output

in Fig. 9. Here we plot in the blue figure the current profileas a function of longitudinal position in the bunch,referenced by the time-of-flight parameter, and in red thenormalized slice emittance. We see that the beam current isvery nearly uniform with a peak of 19 A. The sliceemittance is seen to be relatively uniform itself, with anaverage value in the core nearly equal to the value of theprojected emittance at the exit of the booster linac. Thisfeature is indicative of a highly optimized emittancecompensation process. The only slice parameter of interestwhich is not plotted is the slice energy spread, which has aroughly uniform value of 300 eV across the bunch length.The small value of the energy spread as compared to state-of-the-art FEL injectors can be attributed to two features ofour design: the relatively high accelerating gradient, and theC-band design frequency which naturally allows less timefor space-charge-induced slice energy spread to develop.These results also present the opportunity to discuss the

five-dimensional beam brightness, defined as B5D ¼ 2I=ϵ2nwith I the current and ϵn the normalized emittance. Weshow the slice-dependence of the five-dimensional bright-ness in Figure 10, where it is seen to take an average valueof 2 × 1016 A=m2. This should be compared to the state-of-the-art photoinjectors driving modern XFELs, such as thepresent SwissFEL injector where the 20 A current and200 nm rad emittance yields B5D ≃ 1015 A=m2 [42]. This isa dramatic increase in brightness, an advance that has fewcomparisons in the recent history of high brightnesselectron beam production.

VI. MICROSCOPIC SPACE-CHARGE EFFECTS

A. Short-range Coulomb interactions in photoinjectors

The injector design as presented thus far produces, inaddition to an extremely low emittance, an energy spreadwhich is well below the state-of-the art. Where a standardphotoinjector produces a beamof several keVenergy spread,we have presented here a beam with a final slice energyspread of just a few hundred eV. While in a traditional FELarchitecture this difference has relatively small impact, as theuse of a laser heater downstream would purposefullyincrease the energy spread, we now find applications whichcould not only make use of this exceedingly low energyspread but may also require it. Indeed, the novel UC-XFELdesign presented in [4], does not utilize a laser heater and infact has an upper bound on input energy spread of 10 keV.Additionally, such a source, if operated at lower charge,would be interesting for ultrafast electron diffraction andmicroscopywhere the uncorrelated energy spread presents alimit on achievable resolution [43].In part because the predicted low energy spread of this

source is quite promising, it is also subject to high levels ofscrutiny concerning the validity of the space-charge inter-action model employed in the simulations up to this point.Mesh-based space-charge algorithms inherently elide overthe microscopic physics associated with short-rangeCoulomb interactions which can give rise to phase spacedilution in two different forms. The first, intrabeamscattering (IBS), is a result of the short-range predomi-nantly binary scattering events between neighboring elec-trons which can lead to phase space dilution, primarilythrough an increase in the uncorrelated energy spread of thebeam [34]. The second, disorder-induced heating (DIH),results from an initially spatially disordered beam distri-bution seeking out a pseudocrystalline configuration of

FIG. 9. The time-dependence of the current (blue) and nor-malized transverse emittance (red) are plotted for the FEL injectorcase. FIG. 10. The time-dependence of the five-dimensional bright-

ness, B5D ¼ 2I=ϵ2nx, is plotted for the FEL injector case.


063401-17

lower potential energy, similarly mediated by short-rangeCoulomb interactions [41]. In lowering the potential energyof its charge configuration, the beam increases its uncorre-lated kinetic energy, a fast process (time-scale ω−1

p ) bywhich an increase in both the emittance and the energyspread is realized. These granular effects demand a space-charge algorithm capable of taking into account point-to-point space-charge effects, at least for electrons which arenear each other, while avoiding large numerical errorsfrom close encounters. This may be realized in a fullpoint-to-point space-charge calculation, which is compu-tationally unwieldy, or more practically with a Barnes-Hutapproach [44].Of these two effects, it is expected that IBS presents a

more important limit to the applicability of the high fieldphotoinjector operated with 100 pC charge since it isexpected to yield a contribution to the uncorrelated energyspread on the order of a keV for the beam chargeconsidered [4]. While present injectors can be evaluatedacceptably by either ignoring or estimating the approximateIBS energy spread, as was done in both [45,4], in thepresent design the IBS contribution may be comparable toor even larger than the non-IBS spread. DIH, on the otherhand, has the primary effect of increasing the beamemittance and is typically much smaller than the thermalemittance employed in the 100 pC gun. It can, however,become relatively more important in a design scaled downto the much lower charges relevant to UED and UEM, sincethe thermal emittance scales linearly with the beam size onthe cathode, or the charge as Q1=3, while the space-chargecontribution scales as Q2=3.Analyzing both of these effects is computationally

unwieldy for the 100 pC operating point we have discussedso far, with its associated 625 million electrons. However,both effects are expected to scale with the beam densityNe=σ2xσz, so we can quantify the IBS-induced energyspread in the 100 pC gun by scaling all beam dimensionsdown an order of magnitude while dropping the charge bythree orders of magnitude to 100 fC. This procedurepreserves the space-charge dominated beam dynamics ofthe beam, including emittance compensation, in the rel-evant low-energy region. The resulting beam has 625,000electrons, a number which is feasible to simulate with aBarnes-Hut space-charge algorithm and which is thus ableto capture the essential features of short-range Coulombinteractions.

B. Scaled simulations of IBS energy spread

We have performed the aforementioned scaled simula-tions of the ultrahigh brightness injector by reducing thecharge by three orders of magnitude and all physical beamdimensions by one order of magnitude so as to preserve thebeam charge density which determines both the emittancecompensation dynamics and the strength of the short-rangeCoulomb interactions. The reduction in the transverse size

at emission implies a concomitant reduction in the thermalemittance, which changes linearly with the spot size on thecathode. In fact, the translation of the 100 pC case to the100 fC case is not quite exact because of certain effectswhich do not scale with the beam density: in particular thecontribution to the emittance owed to time-dependent rfeffects and the space-charge field of the image charges inthe cathode. As a result, the emittance compensation profileis not perfectly retrieved at low charge without modifyingother parameters in the injector. Nevertheless, since we areprimarily interested in the effect on the energy spread andwould like tomaintain a proper comparison between the twocases, we will neglect the effects on the emittance compen-sation and consider the energy spread growth which occursonly up to the entrance to the booster linac. This is allowablebecause the dominant source of energy spread growthoccurs when the beam is densest—at the waist ahead ofthe booster—and thus by stopping the simulation at thispoint we will have captured the majority of the IBScontribution in the injector. The scaled simulations are alsoperformed using GPT. In order to properly account for thecathode image charge effect, three space-charge routines areemployed simultaneously: the Barnes-Hut simulationwhichdoes not account for image charges in the cathode, a meshalgorithm including the image charges, and a mesh algo-rithm not including the image charges with an overall minussign applied to the space-charge field. In this way, one cantake into account the image charge field using a meshalgorithm and the real beam space-charge field using aBarnes-Hut algorithm without redundant application of thebeam’s own space-charge field. This is the same proceduredescribed in the GPT manual [46].

1. Scaled simulation results

We have run this scaled simulation and extracted theuncorrelated energy spread of the beam throughout the gunand drift prior to the booster linac, which we display inFig. 11 alongside similar data from another scaled simu-lation using a mesh-based space-charge algorithm. Weobserve that, as expected, the energy spread is increasedby the short-range Coulomb interactions included in theBarnes-Hut simulation. In particular, the correspondingenergy spread growth is most dramatic as the beamapproaches a transverse waist at the entrance to the boosterlinac at z ¼ 1.165 m, as this is the point where the beamdensity is at its largest value. In these simulations, as wewill see in Fig. 13, this waist actually appears closer toz ¼ 1.05 m, hence the peak in the gradient of the uncorre-lated energy spread occurring nearer to this point. When thefull booster linac is included, the energy spread onlyincreases by roughly an additional 100 eV.We may attempt to contextualize this energy spread

growth by comparing it to the theoretical predictions madein [47]. In that work, it was found that the energy spreadgrowth rate satisfied


063401-18

1

σγ

dσγdz

¼ r2eNb

σxσzϵnxσ2γ; ð49Þ

where re ¼ e2=4πϵ0mc2 is the classical electron radius andNb is the number of particles in the bunch. The originalpaper [47] treated the rms beam quantities which appear inthis expression in an averaged sense over the length of theinjector, however one can just as easily include their zdependence. This may be more conveniently written in theform of the derivative of the square of the energy spread,

dσ2γdz

¼ 2r2eNb

σxσzϵnx: ð50Þ

In Fig. 12 we have plotted the Barnes-Hut simulationenergy spread along with the analytic prediction, which wehave obtained by numerically integrating Eq. (50) using thebeam size, beam length, and emittance arrays from the GPTsimulation. We note that although the final energy spread iscomparable between the two cases, the analytic growth rateseems to overestimate the IBS contribution in the drift priorto the waist and underestimates the contribution from thewaist itself.

2. Validation of charge scaling

As previously discussed, short-range Coulomb effectsintuitively scale roughly with the bunch charge density,hence our approach to making this problem tractable. Moreprecisely we see from the theoretical predictions repro-duced in Eq. (50) that the true scaling is not quite with thecharge density but involves the beam angle through theemittance with the term Q=σxσzϵnx. For the sake ofextrapolating the results of these scaled, 100 fC simulations

FIG. 13. The two ratios of interest, the charge density (top) andQ=σxσzϵnx (bottom), are plotted for the 100 pC mesh simulationand the 100 fC Barnes-Hut simulation.

FIG. 12. The uncorrelated energy spread is shown as computedby the analytic formula (orange) and as produced by the Barnes-Hut simulation (blue).

FIG. 11. The uncorrelated energy spread is shown from twodifferent simulations of a scaled 100 fC injector. The Barnes-Hutsimulation (blue) demonstrates the energy spread growth fromshort-range Coulomb effects relative to a baseline case (orange)which utilizes a mesh-based space-charge algorithm.


063401-19

to the original 100 pC gun, we should thus demonstrate thatthis ratio, as well as the beam density, is roughly preservedthroughout the gun when the scaling is performed. Wevalidate this numerical approach in Fig. 13, where thecharge density (top) and the ratio Q=σxσzϵnx (bottom) areplotted for the 100 pC simulation and for the 100 fCBarnes-Hut simulation. As expected from the envelopeequation and emittance compensation theory, there is good,though not perfect, agreement between the two simulationsfor both quantities of interest. Quantitatively, they differfrom each other by no more than a factor of two at thetransverse beam waist where short-range Coulomb inter-actions are strongest. Since the square of the energy spreadhas been shown to be proportional to these ratios, one maythen interpret our 100 fC result as giving an estimate of theIBS contribution to the 100 pC gun energy spread withinroughly 50%. This additional energy spread, of roughly600 eV magnitude, is added in quadrature with the energyspread from the 100 pC simulation.

3. Emittance growth estimate fromdisorder-induced heating

Finally, we will attempt to estimate the magnitude of theexpected emittance growth from DIH in this scaled sce-nario. Associated with disorder induced heating is anincrease in the effective transverse temperature of theelectron beam [41],

kBTDIH ¼ 0.45e2

4πϵ0a; ð51Þ

where a ¼ ð4πnb=3Þ−1=3 is the Wigner-Seitz radius asso-ciated with the electron number density nb at the cathode.This temperature increase is achieved on the timescale of asingle plasma oscillation, similar to the emittance com-pensation dynamics themselves. This corresponds to anincrease in the transverse beam emittance of magnitude

ϵDIH ¼ σx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBTDIH

3mc2

r: ð52Þ

As what will be an overestimate of the DIH effect, let us usethe parameters as they are at the cathode, although in realitythe temperature increase is achieved over a single plasmaperiod and thus samples something closer to the average,lower beam density. For the parameters at the cathode wefind an effective DIH temperature increase of 4.2 meV,resulting in an emittance growth of 0.06 nm rad. Thisimplies a negligible contribution to the beam dynamics,validating our emphasis on IBS effects. Indeed this mag-nitude is completely ignorable relative to the 10’s of nmscale of the 100 pC emittance, however can becomeimportant as the charge is scaled down for potentialUED and UEM applications.

VII. MAGNETIZED OPERATION FORASYMMETRIC EMITTANCE BEAMS

In the following sections we describe an alternativeoperating point for the injector in which the photocathode ismagnetized by an additional solenoid placed behind it. Thebeam generated at the cathode is imbued with canonicalangular momentum: an emittancelike contribution to thetransverse beam dynamics which is physically realized inlinear correlations between the phase space variables x − y0and y − x0. These correlations may be removed downstreamof the injector via skew quadrupole magnets, but only afterthe radial emittance is properly compensated. The resultingbeam can have asymmetric transverse emittances withϵx=ϵy on the scale of 100 while maintaining a four-dimensional emittance, ϵ4D ¼ ffiffiffiffiffiffiffiffi

ϵxϵyp , on the scale of the

ultrahigh brightness FEL injector. Such a beam is attractivefor several applications: at low charge this “flat” beam isideal for injection into a dielectric laser accelerator with aslab geometry. At higher charge, on the order of 1 nC, flatbeams are demanded for a future linear collider in order tomitigate beamsstrahlung at the interaction point. In order tomaximize the versatility of the design and physics wepresent here, we will identify an operating point with100 pC charge which may be scaled up or down accordingto the requirements of the application.

A. Beam requirements for a future linear collider

Linear collider designs require an asymmetric beam at thefinal focus,where the ratio of horizontal to vertical beam sizeis nearly two orders of magnitude [6], to mitigate thenegative effects of the strong beam-beam interaction [7].This in turn implies an emittance ratio ϵx=ϵy ≃ 100, with thesmaller emittance, ϵy, at the 10−8 m rad level. In order to theprovide the needed collider luminosity, there should also bea significant charge in the beam. Traditionally, such phasespace requirements are met by damping rings, whichnaturally produce asymmetric beams. However, such ringsare costly, and it has long been speculated that one mightreplace the electron damping ring with an asymmetricemittance-producing photoinjector.Production of asymmetric beams with a photoinjector is

indeed enabled by strongly magnetizing the photocathode,and subsequently removing the beam’s angular momentumwith a skew-quadrupole array [48]. In the process, one mayproduce a large splitting of the transverse emittances. Aswe shall see below, the conservation laws at play imply thatto reach linear collider-appropriate performance levels, onemust press the state-of-the-art in beam brightness producedby the source. Thus the very high field photoinjector is ofinterest in this application. This approach requires anunderstanding of beam brightness limitations of a magnet-ized beam due the effect of beam angular momentum on theemittance compensation process. Further, one must opti-mally perform the removal of angular momentum through a


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“flat-beam transform.” This second step is discussed first,as it sets the requirements for the first.

B. Flat beam transform of magnetized beams

In principle there are several approaches to producingflat electron beams from photoinjectors. By utilizing ablade array geometry for the photocathode [49] or byshaping the transverse laser profile [50], for example, onecan produce beams with large transverse emittance ratiosdirectly. However, emittance compensation is difficultwhen the beam is not cylindrically symmetric, just onereason for which being that the Larmor rotation of the beamin subsequent solenoid magnets can cause growth of theprojected emittances through unwanted coupling betweentransverse phase space planes. For an azimuthally sym-metric beam this entails no deleterious effects, but for anasymmetric beam this rotation can spoil the beamemittance.Partially on account of these limitations, in this design

we utilize a robust approach based on introducing canoni-cal angular momentum to the beam through immersion ofthe cathode in an axial magnetic field using an externalsolenoid magnet [51]. This technique has the significantadvantage that the beam maintains its cylindrical symmetryup until its emittance is fully compensated; only afterspace-charge effects are strongly diminished is the beampermitted spatial asymmetries in design. Insofar as trackingthe transverse envelope dynamics is concerned, the angularmomentum acts effectively like additional emittance, asnoted above, added in quadrature to the thermal emittanceof the cathode. Unlike the thermal emittance, however, thecanonical angular momentum may be removed by skewquadrupole magnets placed downstream of the injector[48], which introduce the asymmetries mentioned above,and serve to split the transverse emittances. Indeed, in theprocess of removing the cross-correlations in the transversephase space, this transformation leaves the beam with anonunity transverse emittance ratio which can in principlebe quite large, often exceeding 100 [51].We begin the discussion of applying this scheme to the

high field photoinjector by reviewing the basic principles offlat beam generation from a magnetized photocathode. Anelectron released with a nonvanishing axial magnetic fieldat the cathode Bzð0Þ will have a conserved canonicalangular momentum L ¼ eBzð0Þr2=2 such that a bunchof transverse size σr is characterized by a canonical angularmomentum value of L ¼ eBzσ

2r=2. Such a beam is then

characterized by split geometric eigenemittances ϵ� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵ24D þ L2

p� L where ϵ4D ¼ ffiffiffiffiffiffiffiffi

ϵxϵyp is the geometric 4D

emittance and L ¼ L=2pz is the geometric analogue ofthe conserved rms canonical angular momentum. In thelimit that L ≫ ϵ4D the eigenemittances are approximatelyϵþ ¼ 2L and ϵ− ¼ ϵ24D=2L. Thus the emittance ratio isϵþ=ϵ− ¼ ð2L=ϵ4DÞ2. This implies a complicated tradeoff

which goes into maximizing the transverse emittance ratio:a larger axial field on the cathode will increase the ratio,however it will also make compensation more difficult andpotentially increase the effective ϵ4D in the final beamthrough poor compensation. Similarly, both the canonicalangular momentum and the thermal emittance increase withthe spot size on the cathode with the latter scaling linearlyand the former quadratically.

C. Baseline injector performance

The geometry of the injector for flat beam mode issimilar to the ultrahigh brightness mode with the additionof a secondary solenoid placed behind the cathode toprovide an axial magnetic field on the cathode surface. Thissolenoid has a similar engineering philosophy as theoriginal magnet described in Sec. II. We will distinguishbetween this and the existing solenoid by referring to thelatter as the compensating solenoid. Additionally, theinjector is followed by three skew quadrupole magnetswhich perform the flat beam transformation after theemittance is properly compensated.

1. Emittance compensation

We will begin the discussion of the injector performancein the flat beam mode by summarizing the results of theemittance compensation. The relevant parameters for theoperating point are given in Table III and the compensationprofile is plotted in Fig. 14. The on-axis field profile is thesame as in the ultrahigh brightness case, see Fig. 7.Since there are relatively few examples of magnetized

beam compensation, we will take some time to discuss thefeatures of this profile and compare it to our arguments inSec. IV. We note that upon exiting the compensatingsolenoid the beam is converging with an envelope angleσ0x ¼ −620 μrad. Neglecting space-charge defocusing, thiscorresponds to an angular momentum dominated waistbeam size of σx;0 ¼ −L=γσ0x ¼ 250 μm, which is only

TABLE III. The injector parameters relevant to emittancecompensation for magnetized beam operation are listed.


Charge pC 100Laser spot size (precut) μm 151Laser spot size (post-cut) μm 76Injection phase ° 44Laser length ps 5.8Peak cathode field MV=m 240Bucking solenoid field T 0.58Compensation solenoid field T 0.48Compensation solenoid FWHM cm 7.4Compensation solenoid center cm 12.5Booster gradient MV=m 52Booster entrance m 1.6Booster phase ° 90


063401-21

slightly smaller than the waist realized in the simulationindicating that the angular momentum derived defocusingforces are comparable to those derived from space-chargeduring the drift prior to the linac. This is in notable contrastto the FEL operating point, for which an equivalentcalculation using the normalized emittance instead of theangular momentum yields a 7 μm beam size. This is verysmall relative to the 63 μm spot realized in the FEL injectorsimulation, which is indicative of the highly dominantspace-charge defocusing effects in that scenario. Note thatthe implications of this are interesting: the beam sizesassociated with space-charge are naturally much smallerthan those associated with angular momentum when theangular momentum is this large even when the two forcesare strictly equal in magnitude. This justifies a perturbativeapproach to solving for the envelope dynamics even whenangular momentum is not completely dominant.It is worth noting that the booster linac entrance is not

coincident with the location of the first emittance maximumas it usually is and was in the FEL operating mode. Thiswas foreshadowed in Sec. IV—the large angular momen-tum contribution makes it such that the beam becomesangular momentum dominated at a much lower energy.Subsequently, for similar accelerating gradient one mustplace the booster linac closer to the final emittanceminimum. If one tries to inject at the waist the requiredaccelerating gradient will be too low and the beam will notproperly focus. It is worth noting that in the case presentedhere, the emittance is frozen almost exactly at the valuewhen the beam size drops below the constant beam sizesolution associated with the angular-momentum-dominateddynamics σ ¼ ð8=ηÞ1=4ðL=γ0Þ1=2 ≈ 210 μm.We have also included in the design a second 1 m linac

structure located 10 cm downstream of the initial boosterlinac. The emittance compensation process is complete

before the beam enters this additional linac as indicated bythe flat emittance profile between the two linacs. Thesecond linac is placed before the skew quadrupole con-figuration for two purposes. First, the beam energy at theexit of the booster is relatively low at just 55 MeV. Thisrelatively low energy beam is susceptible to space-chargeeffects during the very sensitive skew quadrupole trans-formation; we thus use a second structure operated at themaximally allowed accelerating gradient to increase theenergy to 210 MeV. Furthermore, the second linac providesadditional focusing to the beam to minimize deleteriousnonlinear effects associated with the second-order quadru-pole transfer matrix components.The final normalized radial emittance produced before

the skew quadrupoles is quite good, reaching a minimumvalue of 85 nm rad as shown in Fig. 14. Based on this wemay estimate an optimal transverse emittance ratio ofnearly 500 with γϵþ ¼ 1.9 μm rad and γϵ− ¼ 3.8 nm rad.

2. Skew quadrupole transformation

The second linac is followed immediately by askew quadrupole triplet (SQT). We model the magnetsas 8 cm long with their consecutive edges placed 1 m apart.The first magnet edge is located 4 m downstream of thecathode. The gradients are −14.53 T=m, 9.27 T=m, and−5.36 T=m, respectively. These optimal values wereobtained initially using ELEGANT [52] for fast optimization,then simulated in GPT to account for 3D transverse space-charge effects.The transverse phase spaces after the SQT are plotted on

top of each other in Figure 15. As indicated in the figure,the SQT leaves the normalized transverse emittances splitbetween 4.2 nm rad in x and 1.74 μm rad in y. This

FIG. 14. The emittance and beam size evolution of themagnetized operating point are shown up to the entrance tothe skew quadrupole triplet. FIG. 15. The transverse phase spaces are plotted on top of each

other after the emittances have been split by the skew quadrupoletriplet.


063401-22

amounts to a transverse emittance ratio of 414, which is inexcess of the requirements of the linear collider. This fallsshort of the factor of 500 predicted by our simpletheoretical estimates. We have determined through com-parison of ELEGANT and GPT results that roughly 0.5 nm radof the final smaller emittance is directly derived from acombination of space-charge effects during the skewquadrupole transformation, chromatic effects owing tothe nonlinear longitudinal phase space, and nonlinearitiesin the quadrupole magnetic fields, none of which areaccounted for in the theory employed for the estimate. Ifthis approximately 0.5 nm rad contribution is ignored weretrieve roughly the theoretical prediction of 3.8 nm rad forthe smaller emittance. Further, we show in Fig. 16 the x-yprojection of the beam.

D. Charge scaling

As we saw in the previous sections, the presentedinjector design is highly versatile as one can move betweenan ultrahigh brightness FEL operating point and a mag-netized asymmetric emittance operating point with rela-tively minor adjustments to the gun operating conditions.This versatility has been demonstrated thus far for a 100 pCbeam. 100 pC is ideal for FEL applications, but it may betoo low to be practical in a linear collider scenario and toohigh for DLA. For a linear collider, at least in commonpresent-day designs, one would like to obtain an operatingpoint with a similar emittance ratio but with 1 nC bunchcharge. Here we will discuss the relevant fundamentalphotoinjector scaling laws that govern such an increase tothe bunch charge. We will conclude with a discussion of thecorresponding challenges for scaling down to the single pCscale for DLA.

The standard approach for scaling the charge of aphotoinjector design is to correspondingly scale the beamdimensions at the cathode such that the beam density isunaffected, thereby preserving the plasma frequency whichsets the length scale for emittance compensation. In thisway we may say that the transverse beam size and the beamlength at injection will scale with the cube root of the beamcharge. Recall also that the canonical angular momentumscales as the square of the beam size on the cathode, L ∝ σ2xwhereas the thermal emittance scales linearly with the samequantity. Under the assumption that the final emittance isnearly equal to the thermal emittance, an assumption wewill study in more detail further below, we may concludethat ϵ4D ∝ σx. As a result, the emittance ratio will scale asϵþ=ϵ− ¼ ð2L=ϵ4DÞ2 ∝ σ2x ≃Q2=3. Thus, one might expectthat increasing the bunch charge by an order of magnitudewould in fact improve the final emittance ratio by roughly102=3. Meanwhile, the smaller emittance ϵ− ¼ ϵ24D=2L isinsensitive to the charge scaling in this limit, so the sub-10 nm smaller emittance would also be preserved inaccordance with the requirements for a collider.This analysis holds only under the original assumption

that the final emittance scales with spot size in the sameway as the thermal emittance, which is not necessarily thecase. In particular, lengthening the bunch at the cathodemay enhance the emittance contribution from the rf wave,which is caused by different portions of the beam samplingdifferent phases of the accelerating wave at the exit iristhereby picking up a time-dependent angular kick. Thiseffect is represented by a contribution to the emittance of

ϵrfx ¼ eE0

2mc2σ2xk2rfσ

2z ; ð53Þ

which is added in quadrature to the other contributions tothe total emittance. For the operating conditions in themagnetized photocathode scenario scaled according to theprescription described above, this contribution comes to beabout 750 nm rad, much larger than the sub-100 nmrad thermal emittance. Therefore, one expects that in thiscase the four-dimensional emittance will be dominated bythe rf contribution, leading to the modified scalingϵ4D ∝ σ2xσ

2z ∝ Q4=3. Correspondingly, the emittance ratio

scales as Q−4=3 and the smaller emittance as Q2=3. The newsmaller emittance will still be of order 10 nm rad, howeverthe ratio will be reduced to be of order 10 itself, which isinsufficient for the collider. The most obvious way toresolve this issue is by lengthening the rf wavelength,perhaps by moving instead to S-band (as studied in [14])where the rf frequency is a factor of 2 smaller and thus the rfemittance is reduced by a factor of four assuming all othervariables are held constant. Indeed, such a change reducesthe rf emittance to the 200 nm rad level. This predictioncorresponds to experience; when one is attempting toproduce nC-class high brightness beams, one shouldoperate at an rf wavelength of at least 10 cm.

FIG. 16. The x-y projection of the beam is shown after the skewquadrupole triplet.


063401-23

In the opposite case, one might naturally imagine scalingdown the bunch charge to appeal to DLA applications.In this case rf effects only become weaker, so we mayreturn to the scaling arguments made previously findingϵþ=ϵ− ∝ Q2=3. Thus the emittance ratio is reduced by areduction in the charge, however for the DLA the quantityof higher performance is the smaller emittance ϵ− ¼ϵ24D=2L which is independent of the beam charge in thislimit. The smaller emittance is the more important quantityhere because it determines whether the beam can beproperly matched into the DLA section: the size of thebeam in the larger plane is largely irrelevant. We note thatthe minimal emittance observed in the current injectordesign, 4 nm rad, is consistent with the constraints noted in[53] required for DLA experiments at the UCLA PegasusLab. It is worth noting also, however, that the reduction inthe emittance ratio makes it such that this scaling is onlyworthwhile when the photocathode itself cannot be oper-ated with sufficiently low thermal emittance.

VIII. CONCLUSIONS

We have presented here a versatile, cryogenically-cooledphotoinjector which promises beams of unprecedented six-dimensional brightness. The fundamental innovationallowing this step toward ever brighter beams is the useof normal conducting rf cavities at cryogenic temperatures tolaunch the beam from the cathode at extremely high fields.This work for the first time has considered a feasibleengineering design for all injector components, in theprocess revealing as of yet largely unconsidered physicsconcepts related to the presence of nonresonant spatialharmonics in the rf field. Indeed we have found that theinclusion of these harmonics, demanded by the structure’spower efficiency, serves to enhance the emittance compen-sation by providing stronger focusing on the beam both as itleaves the cathode and as it is accelerated toward anemittance dominated state downstream, yielding an unprec-edented 45 nm rad emittance at 100 pC and 20 A levelcurrent. With the cryogenic nature of the gun comes addi-tional engineering difficulties with the solenoid required tofacilitate emittance compensation, whichwe have addressedwith a design for a cryo-solenoid which can rest in the samecryostat as the gun.In addition to exploring critical new aspects of the concept

of a high-field cryogenic injector, we have presented aunique study of the effects of microscopic collective beamdynamics on the performance of a photoinjector. Ourapproach, which takes advantage of the fundamental prin-ciples driving these short-range Coulomb interactions, hasallowed us to estimate the scale of the energy spread incurredby a beam due to intra-beam scattering in an injector—anestimate which is not just novel but also critical for thefeasibility of an ultracompact x-ray free-electron laser usingcryo-rf accelerators.

We have further shown that the same fundamental injectordesign may be modified by the inclusion of a photocathode-magnetizing solenoid behind the gun to realize beams withunprecedentedly small, asymmetric transverse emittancesand a comparably small four-dimensional emittance. Thisentails a uniquely detailed study of the process of emittancecompensation in the presence of canonical angular momen-tum. The beam demonstrated in these simulations—with100 pC charge, an emittance ratio of 400, and a smalleremittance of 4 nm rad—sheds light on the efficiency ofemittance compensation in this regime and additionallyprovides a first step for developing a scaled beam at a largercharge suitable, for injecting into a linear collider.The experimental realization of this gun is in progress at

the UCLA SAMURAI laboratory, for testing both FEL andlinear collider applications. Before commissioning thisgun, early tests will seek to demonstrate the efficiencygains induced by using a high-field, strongly focusingdistributed coupling linac as the booster for an existinginjector—the UCLA hybrid gun [54]. Simultaneously,UCLA is developing a 0.5 cell C-band test cavity fordedicated study of the properties of beams emitted fromcryogenically cooled photocathode surfaces. This naturallyincludes the possibility of studying cryo-emission in depth,in addition to the behavior of cryogenically cooled cath-odes exposed to high intensity UV lasers. Developing aproper understanding of the details of emission in thisphysical scenario is critical for the small spot size operationwe envision. UCLA has acquired a 5 MWC-band klystron,which is being commissioned and is capable of feeding thishalf-cell structure. With this variety of beam sources soonto be available at the SAMURAI lab, early experimentalwork will also focus on demonstrating the utility of theselinac structures and gun designs in permitting robust FELlasing. The first iterations of this will entail lasing at opticaland EUV wavelengths with a several hundred MeV beam.Such an initial step allows for early investment into theother enabling technologies of the UC-XFEL, such asmeso-scale cryo-undulators and IFEL modulation-basedcompression schemes. These first steps will enable therealization of the full UC-XFEL, producing soft x-rayphotons with a 1 GeV electron beam in just 40 m.

ACKNOWLEDGMENTS

This work was performed with support of the NationalScience Foundation through the Center for Bright Beams,Grant No. PHY-1549132. Support was also obtained fromthe U.S. Department of Energy, Division of High EnergyPhysics, under Contracts No. DE-SC0009914 and No. DE-SC0020409.

APPENDIX A: OPTIMAL π-MODE FIELDPROFILE FOR GRADIENT ENHANCEMENT

The acceleration of ultrarelativistic particles in a standingwave cavity has the interesting property that the rate of


063401-24

energy gain by the particles is sensitive only to thefundamental spatial harmonic, while the walls of the cavityitself are sensitive primarily to the net field strength. Asindicated in previous sections, this implies that the accel-erating gradient observed by an ultrarelativistic particlemay be higher than the peak longitudinal electric field inthe cavity if the field contains higher-order spatial harmoniccontent. This in turn suggests that structures which supporthigh harmonic content will naturally be more efficient atsupporting high gradients, a claim which seems to havebeen validated by the numerical optimizations of [15]which yielded our particular field profile. In this sectionwewill demonstrate that the theoretical limit of this effect isachieved by a perfect square wave field profile, for whichthe accelerating gradient exceeds the expected value by thefactor 4=π.To start we should formally state the problem to be

solved. This is to determine for what valid π-mode fieldprofile is the maximum value of the field a minimumrelative to the strength of the fundamental acceleratingwave. We propose that the answer is a square wave definedaccording to

E0ðxÞ ¼�2=π −π=2 < x < π=2

−2=π π=2 < x < 3π=2: ðA1Þ

Of course, a π-mode field profile is any field profilewhich may be expressed in the form of an odd-frequencycosine series,

EðxÞ ¼X∞n¼0

a2nþ1 cosðð2nþ 1ÞxÞ; ðA2Þ

in which we force the normalization a1 ¼ 1. We will provethat the square wave E0ðxÞ is the answer to the posedproblem by contradiction, by supposing that in fact anotherfield has a smaller maximum than the square wave whenboth have a1 ¼ 1. We will define this field according to itsdifference from the square wave,

EðxÞ ¼ E0ðxÞ þ E1ðxÞ; ðA3Þ

where by necessity E1ðxÞ is itself expressible as an odd-frequency cosine series,

E1ðxÞ ¼X∞n¼0

b2nþ1 cosðð2nþ 1ÞxÞ: ðA4Þ

Since the square wave achieves its maximum value at everylocation in the range ð−π=2; π=2Þ, EðxÞ can only have alower maximum if E1ðxÞ is negative everywhere in thisrange. The reason is simply that if at any point E1ðxÞ ispositive, then EðxÞ will at that point exceed the constantmaximum value of the square wave. Simultaneously how-ever, EðxÞmust maintain a fundamental wave strength of 1.

SinceE0ðxÞ itself already has fundamental coefficient equalto 1, we must have that the corresponding coefficient ofE1ðxÞ vanishes. In other words,

b1 ¼1

π

Z3π=2

−π=2E1ðxÞ cosðxÞdx ¼ 0: ðA5Þ

Since both E1ðxÞ, on account of its form, and cosðxÞ havethe property that fðxþ πÞ ¼ −fðxÞ, this is equivalentlywritten as

b1 ¼2

π

Zπ=2

−π=2E1ðxÞ cosðxÞdx: ðA6Þ

Of course, if E1ðxÞ is negative across the entire integrationrange, then this cannot vanish. Thus E1ðxÞ must becomepositive at some point in this range, thereby granting EðxÞ amaximum which is larger than that of the square wave.With this fact established we may consider its ramifi-

cations for an ideal—under the present consideration—π-mode field profile. The Floquet form of a square waveaccelerating field is

EðzÞ ¼ 2E0

X∞n¼0

ð−1Þn2nþ 1

cosðð2nþ 1ÞkrfzÞ: ðA7Þ

It follows that the maximum of the field profile is related tothe amplitude of the fundamental wave by

Emax ¼ 2E0

X∞n¼0

ð−1Þn2nþ 1

¼ π

2E0: ðA8Þ

The expected accelerating gradient from a purely sinusoidalfield profile is half of the peak field, so we conclude that theaccelerating gradient E0 for an ultrarelativistic particle isE0 ¼ ð4=πÞðEmax=2Þ, implying an enhancement over theequivalent purely sinusoidal accelerating gradient of4=π ≈ 1.27. Stated more explicitly, the square wave fieldprofile achieves a 27% higher accelerating gradient for thesame peak field strength relative to a pure first harmonicwave. In particular, this implies a 27% higher allowedgradient before rf breakdown becomes problematic.

APPENDIX B: ABSENCE OF INVARIANTENVELOPE TYPE SOLUTIONS INMAGNETIZED PHOTOINJECTORS

The invariant envelope is attractive as a particular solutionto the envelope equation in a booster linac largely due to theinvariance of the associated phase space angle with respectto local current. This guarantees that as the beam isaccelerated, first-order correlations in the phase spaceare removed by the time the beam becomes emittance-dominated. Ideally one would like to find an analogoussolution in the presence of angular momentum, however we


063401-25

will show here that no such solution exists. This is not to saythat there are not solutions which facilitate compensation,just that there is no solution with the property that mðzÞ ¼σ0ðz; IÞ=σðz; IÞ is independent of the current, and therefore itis more difficult to guarantee efficient compensation.Proving this statement begins by replacing σ0ðzÞ in the

envelope equation by the trace space angle mðzÞ definedabove. This yields

m0ðzÞ þmðzÞ2 þ γ0

γmðzÞ þ η

8

�γ0

γ

�2

¼ L2

γ2σðz; ζÞ4 þI

2I0γ3σðz; ζÞ2: ðB1Þ

If we ask that the trace space angle not vary with current,then the left-hand side is independent of ζ. As a result wemust also have the right-hand side independent of thecurrent. Let us define

gðzÞ ¼ L2

γ2σðz; ζÞ4 þI

2I0γ3σðz; ζÞ2ðB2Þ

where the function gðzÞ is explicitly independent of thecurrent. This expression can be inverted to write the beamsize in terms of this function,

σðz; ζÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I4I0gðzÞγ3

�1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 8I0gðzÞL2γ4

I

r �s: ðB3Þ

Under our original assumptions, the trace space anglecorresponding to this envelope solution should satisfydm=dI ¼ 0. Enforcing this condition returns a conditionon gðzÞ,

dgdz

¼ −4gγ0

γðB4Þ

This of course can be solved simply

gðzÞ ¼ gð0Þ�γiγ

�4

ðB5Þ

If one plugs this back into the envelope, and subsequentlyinto the envelope equation, there is no solution. Thus, theenvelope equation with angular momentum does not permitsolutions with constant phase space angles throughout thebooster linac.

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