Equilibrium of an arbitrary bunch train in the presenceof multiple resonator wakefields

Robert Warnock *

SLAC National Accelerator Laboratory, Stanford University, Menlo Park, California 94025, USAand Department of Mathematics and Statistics, University of New Mexico,

Albuquerque, New Mexico 87131, USA

(Received 16 November 2020; accepted 19 January 2021; published 24 February 2021)

A higher harmonic cavity (HHC), used to cause bunch lengthening for an increase in the Touscheklifetime, is a feature of several fourth generation synchrotron light sources. The desired bunch lengtheningis complicated by the presence of required gaps in the bunch train. In a recent paper the author andVenturini studied the effect of various fill patterns by calculating the charge densities in the equilibriumstate, through coupled Haïssinski equations. We assumed that the only collective force was from the beamloading (wakefield) of the harmonic cavity in its lowest mode. The present paper improves the notation andorganization of the equations so as to allow an easy inclusion of multiple resonator wakefields. This allowsone to study the effects of beam loading of the main accelerating cavity, higher order modes of the cavities,and short range geometric wakes represented by low-Q resonators. As an example these effects areexplored for ALS-U. The compensation of the induced voltage in the main cavity, achieved in practice by afeedback system, is modeled by adjustment of the generator voltage through a new iterative scheme. Exceptin the case of a complete fill, the compensated main cavity beam loading has a substantial effect on thebunch profiles and the Touschek lifetimes. A Q ¼ 6 resonator, approximating the effect of a realistic shortrange wake, is also consequential for the bunch forms.

DOI: 10.1103/PhysRevAccelBeams.24.024401

I. INTRODUCTION

This is a sequel to Ref. [1], in which we explored theaction of a higher harmonic cavity (HHC), a standardcomponent of 4th generation synchrotron light sources,employed to lengthen the bunch and reduce the effect ofTouschek scattering. In that work we introduced an effectivescheme to compute the equilibrium state of charge densitiesin an arbitrary bunch train. The train is allowed to havearbitrary gaps and bunch charges. We chose the simplestpossible physical model, in which the only induced voltage(wakefield) is due to the lowest mode of the HHC.WewriteVr3 for this voltage, the notation designating “resonator, 3rdharmonic”. We recognized, however, that excitation of themain accelerating cavity (MC) by the bunch train producesan induced voltage Vr1 of comparable magnitude, the effectdescribed as beam loading. Our excuse for omittingVr1 wasthat in practice it is largely cancelled by adjusting the rfgenerator voltageVg through a feedback system. The sum of

Vr1 and Vg should closely approximate Vrf, the desiredaccelerating voltage.In real machines there are always gaps in the bunch train,

and that leads to varying bunch profiles and centroiddisplacements along the train. At first sight this wouldsuggest that Vr1 would be different for different bunches, sothat compensation could only be partial, perhaps onlymanifest in some average sense. On the contrary, we shallcalculate an equilibrium state in which the compensation isessentially perfect for all bunches. This happens by anadjustment of the charge densities of all bunches, to newforms that sometimes differ substantially from those withouttheMC. The adjustment is achieved automatically through anew algorithm presented here. This iterative procedureminimizes a mean square deviation of Vr1 þ Vg fromVrf, summedover all bunches, as a function of twogeneratorparameters, which are equivalent to amplitude and phase.It is not clear that this is a faithful model of the feedback

mechanism, which could conceivably amount to a weakerconstraint on the bunch profiles. Nevertheless, this studyclarifies the mathematical structure of the problem, andappears to be a worthwhile preliminary to a full time-dependent model of the system including a realisticdescription of feedback.Beside the main cavity, we also assess the role of the

short range wakefield from geometric aberrations in the

*warnock@slac.stanford.edu

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW ACCELERATORS AND BEAMS 24, 024401 (2021)

2469-9888=21=24(2)=024401(13) 024401-1 Published by the American Physical Society

vacuum chamber, and higher order modes in the HHC.These effects are added with the help of improvements innotation and organization of the equations.Extensive numerical results are reported with parameters

for ALS-U. For consistency with the previous work theparameters chosen are partially out of date as the machinedesign stands at present, but that will not greatly affect ourprincipal conclusions. Although the qualitative picture ofthe model with HHC alone is still in place, there are largequantitative changes. Even then, we underestimate thefull effects, because we can only get convergence of ouriterative method when the current is a few percent less thanthe design current.In Section II we briefly recall our previous algorithm

for solving the coupled Haïssinski equations. Section IIIintroduces the improved notation and organization whichallows an easy inclusion of multiple resonator wakefields.Section IV enlarges the system of equations to provide acalculation of the diagonal terms in the potential, thusovercoming a limitation of the previous formulation.Section V describes the method for determining thegenerator parameters so as to compensate the inducedvoltage in the main cavity. Section VI, with severalsubsections, reports numerical results for the case ofALS-U [2,3], always making comparisons to results withonly the HHC in place. Subsection VI A treats the case of acomplete fill, illustrating the compensation of the maincavity in the simplest instance. Subsection VI B considers apartial fill with distributed gaps, as proposed for themachine. SubsectionVI C is concernedwith over-stretchingby reduction of the HHC detuning. Subsection VI Dexplores the effect of the short range wakefield with arealistic wake potential. Subsection VI E checks theeffect of the principal higher order mode of the HHC.Subsection VI F presents our closest approach to arealistic model, including the harmonic cavity, the compen-sated main cavity, and the short range wake, altogether.Subsection VI G examines the effect of the main cavitywhen there is only a single gap in the bunch train. SectionVIIreviews our conclusions and possibilities for further work.Appendix derives the expression for the diagonal terms inthe potential, for resonators of arbitrary Q.

II. SUMMARY OF METHOD TO COMPUTE THEEQUILIBRIUM CHARGE DENSITIES

In [1] we derived a set of equations to determine theequilibrium charge densities of nb bunches, which may bestated succinctly as follows:

Fðρ; IÞ ¼ 0: ð1Þ

Here I is the average current and ρ is a vector with 2nb realcomponents, consisting of the real and imaginary parts ofρiðkr3Þ, where kr3 is the wave number of the lowestresonant mode of the 3rd harmonic cavity. These quantities

are defined in terms of the beam frame charge densitiesρiðzÞ, normalized to 1 on its region of support ½−Σ;Σ�, as

ρiðkÞ ¼1

2π

ZΣ

−Σexpð−ikzð1þ i=2QÞÞρiðzÞdz; ð2Þ

whereQ is the quality factor of the cavity. The vector in (1)is arranged as follows:

ρ ¼ ½Reρ1ðkr3Þ;…;Reρnbðkr3Þ; Imρ1ðkr3Þ;…; Imρnbðkr3Þ�:ð3Þ

Accordingly, F in (1) is a real vector with 2nb components,so that we have 2nb nonlinear algebraic equations in 2nbunknowns, depending on the parameter I.For the high Q of a typical HHC the quantity (2) is very

close to the Fourier transform, but we have persistentlywritten all equations for general Q for later applicationsinvolving low-Q resonators.In (1) the diagonal terms of the induced voltage have

been dropped, i.e., the effects on a bunch of its ownexcitation of the cavity. This omission is justified for thetypical high Q of an HHC. Our method to handle thediagonal terms in the general case is introduced in Sec. IV.A solution ρ of (1) determines the charge densities by the

formula of Eq. (50) in [1],

ρiðziÞ ¼1

Aiexp½−μUiðziÞ�; ð4Þ

where Ui is the potential felt by the ith bunch, defined inEq. (51) of [1]. Here μ and Ai are constants, and zi is thebeam frame longitudinal coordinate of the ith bunch. Thepotential Ui depends on all components of ρ, on the meanenergy loss per turn U0, and on the parameters of theapplied voltage Vrf which we write as

VrfðzÞ ¼ V1 sinðk1zþ ϕ1Þ¼ V1½cosϕ1 sinðk1zÞ þ sinϕ1 cosðk1zÞ�: ð5Þ

We solve (1) by the matrix version of Newton’s iteration,defined in (67) of [1]. We begin at small current I, taking allcomponents of the first guess for ρ to be the transform (2) ofa Gaussian with the natural bunch length. We then continuestepwise to the desired current, making a linear extrapo-lation in current to provide a starting guess for the nextNewton iteration at incremented current. The extrapolationis accomplished by solving for ∂ρ=∂I from the I-derivativeof (1):

∂F∂ρ

∂ρ∂I þ

∂F∂I ¼ 0: ð6Þ

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III. FORMALISM FOR MULTIPLE RESONATORS

The scheme allows the inclusion of any number ofresonator wakefields, but to do that conveniently requires

some care in notation and organization of the equations.With nr resonators there are 2nbnr ¼ nu unknowns, whichwe assemble in one long vector ρ:

ρ ¼ ½ρðkÞ; k ¼ 1;…; nu�¼ ½Reρ1ðkr1Þ;…;Reρnbðkr1Þ; Imρ1ðkr1Þ;…; Imρ1ðkr1Þ;…;

Reρ1ðkr;nrÞ;…;Reρnbðkr;nrÞ;…; Imρ1ðkr;nrÞ;…; Imρnbðkr;nrÞ�: ð7Þ

Here kr;n is the resonant wave number of the nth resonator,and the subscript of ρ denotes as usual the bunch number.To identify the bunch number and the resonator number

for the kth component of the vector, we define two indexmaps: ιðkÞ which gives the bunch number and rðkÞ whichgives the resonator number. Namely,

ιðkÞ ¼�mod ðk; nbÞ if mod ðk; nbÞ ≠ 0

nb if mod ðk; nbÞ ¼ 0

�ð8Þ

rðkÞ ¼�

k2nb

�: ð9Þ

Here dxe, the ceiling of x, is the least integer greater than orequal to x. We also need two projection operators: PreðkÞwhich is equal to 1 if k corresponds to a Reρ and is zerootherwise, and PimðkÞ which is equal to 1 if k correspondsto a Imρ and is zero otherwise. These are expressed in termsof the ceiling of k=nb as follows:

PreðkÞ ¼1

2½1 − ð−1Þdk=nbe�;

PimðkÞ ¼1

2½1þ ð−1Þdk=nbe�: ð10Þ

The potential UjðzÞ for bunch j, generalizing Eq. (51) of[1] to allow nr resonators, is stated as

UjðzÞ ¼eV1

k1½x1 cosðk1zÞ − x2 sinðk1zÞ − x1� þU0z ð11Þ

þXnrn¼1

UdjnðzÞþ

Xnuk¼1

MðzÞj;kρk; j¼1;…;nb; −Σ≤ z≤Σ:

ð12ÞThe first term in (11) is −e times the integral of the appliedvoltage, now called the generator voltage and written as

VgðzÞ ¼ V1½x1 sinðk1zÞ þ x2 cosðk1zÞ�: ð13ÞAt x1 ¼ cosϕ1, x2 ¼ sinϕ1 this reduces to the desired Vrf

of (5). In an amplitude-phase representation we have

VgðzÞ ¼ V1 sinðk1zþ ϕ1Þ; V1 ¼ ðx21 þ x22Þ1=2V1;

ϕ1 ¼ tan−1ðx2=x1Þ: ð14Þ

The first term in (12) represents the diagonal contributions,the effect on bunch j of its own excitation of the resonators,as opposed to excitation by the other bunches which isdescribed by the second term. By writing the latter as asimple matrix-vector product we greatly simplify thecalculation of the Jacobian of the system, making itformally the same for any number of resonators.Referring to Eqs. (28), (51), (55), (56), (57), (58) of [1],

we can write down the matrix elements MðzÞi;k in thesecond term of (12). For this we introduce a notationappropriate for labeling by the index k of (7). Functionsof k, defined via the index maps, are labeled with a tilde:

kr;k ¼ kr;rðkÞ;

ξk ¼ ξιðkÞ; Ak ¼ AιðkÞ

Rsk ¼ Rs;rðkÞ; Qk ¼ QrðkÞ;

ηk ¼ ηrðkÞ; ψk ¼ ψ rðkÞ;

ϕj;k ¼ kr;k½ðmιðkÞ −mjÞλ1 þ θj−1;ιðkÞC�;σj;kðzÞ ¼ Sðkr;kz; Qk; ϕj;k þ ψkÞ;γj;kðzÞ ¼ Cðkr;kz; Qk; ϕj;k þ ψkÞ; ð15Þ

where

Sðkrz;Q;ϕÞ¼ 1

1þð1=2QÞ2�expð−krz=2QÞ

�sinðkrzþϕÞ

−1

2QcosðkrzþϕÞ

�z

0

;

Cðkrz;Q;ϕÞ¼ 1

1þð1=2QÞ2�expð−krz=2QÞ

�cosðkrzþϕÞ

þ 1

2QsinðkrzþϕÞ

�z

0

: ð16Þ

The result for the matrix from (51) and (57) of [1] is seen tobe (noting that ωr=kr ¼ c)

MðzÞj;k¼ 2πce2NηjRsj

Qjð1−δj;ιðkÞÞξk

×expð−ϕj;k=2QkÞ½PreðkÞσj;kðzÞþPimðkÞγj;kðzÞ�:ð17Þ

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In the present notation the system of coupled Haïssinski equations, generalizing (66) of [1], takes the form

FjðρÞ¼ Ajρj−1

2π

ZΣ

−Σ½PreðkÞcosðkr;jζÞ−PimðkÞsinðkr;jζÞ� · exp½kr;jζ=2Qj−μUιðjÞðζÞ�dζ¼ 0; j¼ 1;…;nu: ð18Þ

The normalization integral appearing in the first term is

Aj ¼Z

Σ

−Σexp½−μUιðjÞðζÞ�dζ: ð19Þ

We require the Jacobian matrix ½∂Fj=∂ρk� for the solution of (18) by Newton’s method, assuming that the diagonal termsare fixed. This is found immediately from (12), (18), and (19) as

∂Fj

∂ρk ¼ Ajδj;k − μ

ZΣ

−Σexp½−μUιðjÞðζÞ�MðζÞj;k ·

�ρj −

1

2π½PreðkÞ cosðkr;jζÞ − PimðkÞ sinðkr;jζÞ� exp½kr;jζ=2Qj�

dζ: ð20Þ

The compact expressions in (17), (18), and (20) are quite convenient for coding, and lead to a short program to solve theHaïssinski equationswith anynumberof resonators.Forζ atnpmeshpointszi used in the integralswehave thearrayMði; j; kÞ ¼MðziÞj;k of manageable dimension np × nb × nu which can be computed and stored at the top, outside the Newton iteration.For the work of the following section we also need the induced voltage from the main cavity, which we designate as the

first resonator in the list (n ¼ 1). For the jth bunch this takes the form

Vr1jðzÞ ¼ −2πceNkr1Rs1η1

Q1

�X2nbk¼1

ð1 − δj;ιðkÞÞξk expð−ðkr;kzþ ϕj;kÞ=2QkÞ

× ðPreðkÞ cosðkr;kzþ ϕj;k þ ψkÞ − PimðkÞ sinðkr;kzþ ϕi;k þ ψkÞÞρk þ vd1jðzÞ: ð21Þ

The diagonal term vd1j can be evaluated in terms of integralsderived in Appendix.

IV. THE FULL SYSTEM OF EQUATIONSWITH DIAGONAL TERMS

Through (18) we have a system of nu algebraicequations for determination of ρ, provided that the diagonalterms in Ui are given. The latter are functionals of thecharge densities ρiðziÞ, from which it follows that (18) canbe stated in vector notation as

ρ ¼ Aðρ; ρ; IÞ: ð22ÞOn the other hand, the ρiðziÞ are determined in turn assolutions of integral equations provided that ρ is given. Theintegral equations are like normal single-bunch Haïssinskiequations, but with a background potential determinedby ρ, namely

ρiðziÞ ¼1

Aiexp½−μUiðzi;ρi; ρÞ�; i¼ 1;…; nb: ð23Þ

In vector notation

ρ ¼ Bðρ; ρ; IÞ:: ð24ÞThe potential Ui depends on the ρi through its diagonalterms, in the first sum in (12). Our procedure will be tointerleave the solution of (22) at fixed ρ, by the usualNewton method, with the solution of (24) at fixed ρ. If this

algorithm converges we shall have consistency betweenρ and ρ and a solution of the full system.It turns out, most fortunately, that the solution of (24) is

obtained by plain iteration as would be applied to acontraction mapping,

ρðnþ1Þ ¼ BðρðnÞ; ρ; IÞ: ð25ÞIn our application this usually converges to adequateaccuracy in just one step, or three at most, and takesnegligible time.This scheme based on (22) and (24) is used in all

calculations reported below. It replaces the method used in[1], which was to evaluate the diagonal terms from thevalue of ρ from the previous Newton iterate. That worksonly for high-Q resonators, so is not adequate for handlingthe short range machine wake.

V. ALGORITHM TO ADJUST THE GENERATORPARAMETERS ðx1; x2Þ

We wish to choose ðx1; x2Þ so as to minimize, in somesense, the difference

VrfðziÞ − Vgðzi; x1; x2Þ − Vr1iðzi; x1; x2Þ; ð26Þfor all i ¼ 1;…; nb. A reasonable and convenient choice foran objective function to minimize is the sum of the squaredL2

norms of the quantities (26). With a normalizing factor tomake it dimensionless and of convenient magnitude that is

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fðx1; x2Þ ¼1

2ΣV21

Xnbi¼1

ZΣ

−Σ½V1ðcosϕ1 − x1Þ sinðk1zÞ þ V1ðsinϕ1 − x2Þ cosðk1zÞ − Vr1iðz; x1; x2Þ�2dz: ð27Þ

The region of integration ½−Σ;Σ� is the same as that used inthe definition of the potential Ui.Note that the minimum of f cannot be strictly zero,

since Vr1 is sinusoidal with wave number kr1, whereas theother terms are sinusoidal with a slightly different wavenumber k1.Let us adopt the vector notation x ¼ ðx1; x2Þ with norm

jxj ¼ jx1j þ jx2j. The equations to solve now depend on x,having the form

Fðρ; I; xÞ ¼ 0: ð28Þ

To avoid notational clutter we suppress reference to thediagonal terms, leaving it understood that a solution of (28)for ρ actually involves the scheme of the previous session.As usual we solve for ρ, for an increasing sequence ofI-values. The scheme will be to minimize fðxÞ at each I,thus providing a new x ¼ arg min f to be used at the nextvalue of I. As will now be explained, the minimization willalso be done iteratively, so that we have an x-iterationembedded in the ρ-iteration.We wish to zero ∇xF, which is to find x to solve the

equations

Xnbi¼1

ZΣ

−Σ½V1ðcosϕ1 − x1Þ sinðk1zÞ

þ V1ðsinϕ1 − x2Þ cosðk1zÞ − Vr1iðz; xÞ�

×

�V1 sinðk1zÞ þ ∂x1Vr1iðz; xÞV1 cosðk1zÞ þ ∂x2Vr1iðz; xÞ

dz ¼

�0

0

: ð29Þ

To solve (29) a first thought might be to apply Newton’smethod, starting at some low current and choosing the zerocurrent solution ðcosϕ1; sinϕ1Þ as the first guess. Thiswould be awkward, however, since it would involve thesecond derivatives of Vr1i with respect to ðx1; x2Þ. The firstderivatives must already be done by an expensive numeri-cal differentiation, and the second numerical derivativewould be error prone and even more expensive. Instead, letus assume that we have a first guess ðx10; x20Þ and supposethat in a small neighborhood of that point the firstderivatives of Vr1i can be regarded as constant. Thensecond derivatives are zero and the first-order Taylorexpansion of Vr1i gives two linear equations to solve forðx1; x2Þ, namely

�a11 a12a21 a22

�x1x2

¼

�b1b2

; ð30Þ

where

a11¼Xi

Zα1iðz;x0Þ2dz; a22¼

Xi

Zα2iðz;x0Þ2dz;

a12¼ a21¼Xi

Zα1iðz;x0Þα2iðz;x0Þdz;

b1¼Xi

Zα1iðz;x0Þβiðz;x0Þdz;

b2¼Xi

Zα2iðz;x0Þβiðz;x0Þdz; ð31Þ

with

α1iðz; x0Þ ¼ V1 sinðk1zÞ þ ∂x1Vr1iðz; x0Þ;α2iðz; x0Þ ¼ V1 cosðk1zÞ þ ∂x2Vr1iðz; x0Þ;βiðz; x0Þ ¼ −Vr1iðz; x0Þ þ∇xVr1iðz; x0Þ · x0

þ V1 sinðk1zþ ϕ1Þ ð32ÞBy (30) we have an update x0 → x which establishes the

pattern of the general iterate xðkÞ → xðkþ1Þ. This will becarried to convergence in the sense jxðkþ1Þ − xðkÞj < ϵx,with a suitable ϵx to be determined by experiment. Eachiterate requires a value for Vr1i and for ∇xVr1i, which wecompute numerically by a divided difference,

∂Vr1i

∂x1 ðz; xÞ ≈ Vr1iðz; x1 þ Δx; x2Þ − Vr1iðz; x1; x2ÞΔx

: ð33Þ

Thus one x-iteration requires three ρ-iterations to providethe necessary values of Vr1i (which are constructed from ρ).The first ρ iteration to find Vr1iðz; x1; x2Þ produces a ρwhich is a very good guess to start the remaining twoiterations to make the derivatives, which then convergequickly.The choice of Δx in (33) requires a compromise between

accuracy and avoiding round-off error. We found that Δx ¼10−4 was widely satisfactory, whereas success with smallervalues depended on the circumstances.

VI. NUMERICAL RESULTS WITH ANDWITHOUT THE MAIN CAVITY

As in [1] we illustrate with parameters for ALS-U[2,3], the forthcoming Advanced Light Source Upgrade.Although the machine design is not yet final, one provi-sional set of parameters for our main cavity (actually theeffect of two cavities together) is as follows:

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Rs¼ 0.8259MΩ; Q¼ 3486;

δf¼ fr1−f1 ¼−82.54 kHz ð34ÞHere the shunt impedanceRs and quality factorQ are loadedvalues, the unloaded values divided by 1þ β, with couplingparameter β ¼ 7.233. We take these parameters for the maincavity, otherwise keeping the same parameters as in [1],Table I. Thuswe takeU0 ¼ 217 keV, even though a value of330 keV may be contemplated for the set (34).

A. Complete fill

We first take the case of a complete fill, thusnb ¼ h ¼ 328. The average current is to be 500 mA, whichwe reach in 8 steps starting from 200 mA. The CPU time is15 minutes, rather than 20 seconds for the calculationwithout the main cavity. The increase is mostly due to amuch slower convergence of the ρ-iteration, the x-iterationbeing a minor factor in CPU time. To save time we gave ϵxthe rather large value of 0.05, but then made a refinement toϵx ¼ 10−6 at the final current, in an extra 2 minutes. Thesteepness of the objective function fðx1; x2Þ of (27) isextraordinary, having values around 104 in the sequencewith ϵx ¼ 0.05 while falling to a value close to 1 after therefinement. An interesting question is how this steepnesswould be reflected in a feedback system.The result for the charge density, shown in the blue curve

of Fig. 1, is quite close to the result without the main cavity,shown in red. It should be emphasized that there is noexplicit constraint requiring all bunches to be the same. Wehave computed 328 bunches separately, and have found thatthey all come out to be the same. This constitutes a goodcheck on the correctness of the equations and the code.In Fig. 2 we show the compensation mechanism. The

sum of the generator voltage Vg and the induced voltageVr1 from the main cavity is the orange curve. The latter

deviates from the desired effective voltage Vrf by less than2%, as is seen Fig. 3.The phasor of the generator voltage moves closer to π=2

and its magnitudeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix21 þ x22

pincreases from 1 to 1.0245, in

comparison to the phasor of Vrf. The corresponding valuesof ðx1; x2Þ areðx10; x20Þ ¼ ðcosϕ1; sinϕ1Þ ¼ ð−0.93231;0.36167Þ→ ðx1; x2Þ ¼ ð−0.29098;0.98018Þ: ð35Þ

B. Partial fill C2 with distributed gaps

Next we take a partial fill with distributed gaps, labeledas fill C2; see Sec. XIII-C of [1]. There are 284 bunches in11 trains, with 4 empty buckets between trains. There are 9trains of 26 and 2 of 25, with the latter positioned atopposite sides of the ring. All bunches have the same

FIG. 1. Charge density for complete fill at 500 mA, withcompensated main cavity (blue) and without main cavity (red). FIG. 2. MC induced voltage Vr1, generator voltage Vg and

their sum.

FIG. 3. Relative deviation of Vr1 þ Vg from Vrf .

ROBERT WARNOCK PHYS. REV. ACCEL. BEAMS 24, 024401 (2021)

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charge. As in the preceding example we start the calculationat low average current and advance in steps trying to reachthe desired 500 mA. The convergence of iterations is at firstsimilar to that of the preceding case, but begins to falteraround 430 mA average current, at which point theconvergence of the ρ-iteration becomes problematic. Bytaking smaller and smaller steps in current we can reach496 mA, but beyond that point the Jacobian matrix of thesystem appears to approach a singularity, as is indicated byits estimated condition number having a precipitousincrease, from 700 at the last good solution to 2900 at aslightly higher current. Nevertheless, the x-iterations con-tinue to converge as long as the ρ-iterations do. In thefollowing, graphs are plotted for the maximum achievablecurrent, stated in figure captions.Now the plots of Vg and Vr1 and their sum look exactly

the same as in Fig. 2, for every bunch. The minimization of

fðx1; x2Þ has caused the bunch forms to rearrange them-selves so that the compensation is essentially perfect forevery bunch. The deviation of Vr1 þ Vg from Vrf, scarcelyvisible on the scale of Fig. 2, varies from bunch to bunch,but is still less than 3% for all bunches.Figure 4 shows 9 bunch profiles in one train, to be

compared with the corresponding results without the maincavity in Fig. 5. The main cavity causes considerably morebunch distortion along the train, and also a bigger variationin the rms bunch lengths, as is seen in Fig. 6. The plotsshow the ratio of bunch length to the natural bunch length.The head of the train is on the right, with the highest bunchnumber.

FIG. 4. Charge densities in a train of 26, surrounded by gaps of4 buckets, fill C2, MC beam loading included, Iav ¼ 496 mA.

FIG. 5. Charge densities in a train of 26, surrounded by gaps of4 buckets, fill C2, MC beam loading omitted, Iav ¼ 496 mA.

FIG. 6. Bunch length increases in a train of 26, surrounded bygaps of 4 buckets, with main cavity beam loading (blue) andwithout (red). Iav ¼ 496 mA. The plot is the ratio of bunch lengthσ to the natural bunch length σ0.

FIG. 7. Centroids hzi in a train of 26, surrounded by gapsof 4 buckets, with cavity beam loading (blue) and without (red).Iav ¼ 496 mA.

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The corresponding results for the bunch centroids is seenin Figs. 7. Again the deviation from the case without themain cavity is quite substantial.The main point of practical interest is the increase in

Touschek lifetime achieved through the bunch stretchingcaused by the HHC. Again, the MC has a sizeable effect inreducing the lifetime and in causing a larger variation alonga train. This is shown in Fig. 8 which gives the ratio of thelifetime τ to the lifetime τ0 without the MC.We next consider the same fill pattern with 11 trains, but

with a taper in the bunch charges putting more charge at theends, according to a power law as shown in Fig. 15 of [1].This is an example of invoking guard bunches to reduce theeffect of gaps. As is seen in Figs. 9 and 10, the guardedinner bunches, which resemble that of the complete fill, arelittle affected by the MC. The strong asymmetry between

the front and back of the train is perhaps surprising, but itshould be noticed that Fig. 10 already shows an appreciablefront-back asymmetry. The strong amplification of thisasymmetry by the MC is in line with its big effects seengenerally.

C. Decrease of HHC detuning for overstretching

There is practical interest in the possibility of over-stretching for an additional increase in the Touschek life-time. This entails a decrease in the detuning of the HHC,which produces a larger r.m.s. bunch length but a bunchprofile with a dip in the middle, thus a double peak. In ourcase a decrease from df ¼ 250.2 kHz to df ¼ 235 kHzproduces a double peak in the model without the MC at fullcurrent, as is seen in Fig. 4 of [1].Wewould like to knowhowthis setup looks with the compensated main cavity in play.Not surprisingly, the convergence of our iterative solutionbreaks down at a lower current than in the case of the normaldetuning; the stronger the bunch distortions the poorer theconvergence. With df ¼ 235 kHz and the MC we can onlyreach 474.5 mA, which is not enough to see a double peak.Nevertheless it is useful to compare the result at that currentwith the result in absence of the MC, as displayed for9 bunches in a train of 27 in Figs. 11 and 12.Even at a current significantly less that the 500 mA

design current the distortion due to the main cavity is quitelarge, which leads to the conclusion that the main cavitymust be included in a realistic simulation of over-stretching.

D. Effect of the short range wakefield

The short range wakefield from various unavoidablecorrugations in the vacuum chamber retains importance inthe latest storage rings, in spite of the best efforts to reduceit. Since it can cause substantial bunch distortion in the

FIG. 8. Touschek lifetime increase along a train, with compen-sated MC (blue) and without (red). Iav ¼ 496 mA.

FIG. 9. Case of tapered bunch charges, MC beam loadingincluded, Iav ¼ 496 mA.

FIG. 10. Case of tapered bunch charges, MC beam loadingomitted, Iav ¼ 496 mA.

ROBERT WARNOCK PHYS. REV. ACCEL. BEAMS 24, 024401 (2021)

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absence of an HHC, we would like to know how much itaffects the operation of the HHC. A result for thelongitudinal wake potential at ALS-U, from a detailedcomputation by Dan Wang [4], is shown in Fig. 13. Thecorresponding impedance,

ZðfÞ ¼ 1

c

Z∞

−∞e−ikzWðzÞdz; f ¼ kc=2π; ð36Þ

is plotted in Fig. 14.In Ref. [1] we suggested that a low-Q resonator wake

could be treated on the same footing as the high-Qresonators, and for that reason we wrote all equationsfor a general value of Q. We recognized, however, that thediagonal term in the potential would now be dominant,while being nearly negligible in the high-Q case. It couldnot be treated by the method used in [1], but is easilyhandled by the presently adopted method of Sec. IV.

For the equilibrium state, the impedance at f > 20 GHzis irrelevant, even though it could have a role out ofequilibrium. This assertion follows from the fact that thefrequency spectrum of our calculated charge densities neverextends beyond 15 GHZ, no matter which wakefields areincluded. Consequently, a reasonable step is to concentrateon the first big peak at 11.5 GHz. The wake potential inour equations (defined in (19) of [1]) is based on animpedance as follows, which is of Lorentzian form withhalf-width Γ=2:

ZðfÞ¼ iRsΓ2

�1

f−frþ iΓ=2þ 1

fþfrþ iΓ=2

¼Zð−fÞ�;

Γ=2¼ fr=2Q: ð37Þ

Figures 15 and 16 show a fit to (37) with parameters asfollows:

FIG. 12. Fill C2 with HHC only, detuning df ¼ 235 kHz,Iav ¼ 474.5 mA.

FIG. 13. Wake potential (pseudo—Green function) for theALS-U storage ring, computed with a 1 mm driving bunch.

FIG. 14. Longitudinal impedance ZðfÞ for ALS-U.

FIG. 11. Fill C2 with HHCþMC, detuning df ¼ 235 kHz,Iav ¼ 474.5 mA.

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fr ¼ 11.549 GHz; Rs ¼ 5730 Ω; Q ¼ 6: ð38Þ

The fit is rough in the imaginary part, but probably goodenough to estimate the magnitude of the effect of the shortrange wake.Discussions of low-Q resonator models in the literature

usually invoke the impedance of an LRC circuit, ZðfÞ ¼R=ð1þ iQðfr=f − f=frÞÞ, often with Q near 1. As isillustrated in Figs. 15 and 16, in our case with Q ¼ 6 theLRC model does not give a better fit than the simplerLorentzian, except for enforcing Zð0Þ ¼ 0. At the expenseof some complication our equations could be modified toaccommodate the LRC form, but that appears to beunnecessary, at least in the present example.Henceforth, the impedance from (37) and (38) will be

referred to as SR (short range). Taking first a complete fill,and including just the HHC and SR, we get the resultof Fig. 17.

Next we consider the partial fill C2 with distributed gapsas treated in the previous section. Figures 18 and 19 showthe results for HHCþ SR and HHC alone. As expected, theeffects of SR are more pronounced in the partial fill than inthe complete fill. Correspondingly, the maximum currentachieved is 472.6 mA. As in previous cases we expect asubstantially larger effect at the design current of 500 mA.

E. Higher order mode (HOM) of theharmonic cavity

At the present stage of design the most prominentlongitudinal HOM of the HHC for ALS-U is a TM011mode with the following parameters [5]:

Rs ¼ 3000 Ω; Q ¼ 80; fr ¼ 2.29 GHz ð39Þ

FIG. 15. Fit of ReZ to Lorentzian and LRC circuit formulas.

FIG. 16. Fit of ImZ to Lorentzian and LRC circuit formulas.

FIG. 17. Charge density for a complete fill, with HHC plus thefirst peak in the short range impedance (blue), and with HHCalone (red). Iav ¼ 500 mA.

FIG. 18. Fill C2, HHCþ SR, Iav ¼ 476.2 mA.

ROBERT WARNOCK PHYS. REV. ACCEL. BEAMS 24, 024401 (2021)

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A calculation for fill C2 with the HHC and this HOM gavethe result of Fig. 20. The effect of the HOM on the chargedensities is less than 2%, in a small shift at the top of thedistributions.At least for the equilibrium state in ALS-U, it appears

that the HOM can be neglected. The role of HOM’s inlongitudinal coupled-bunch instabilities is discussedin Ref. [6].

F. The full model: HHC+ MC+SR

We are now prepared to include the harmonic cavity, thecompensated main cavity, and the short range wake,altogether. The convergence of the Newton sequencesuffers even more than in the previous cases, and thecontinuation in current reaches only Iav ¼ 471.9 mA.Effects seen at this current must severely underestimatewhat can be expected at 500 mA, because of the strong

variation near the design current that we have observed inevery case.For fill C2 we see the charge densities in Figs. 21 and 22.

G. The case of a single gap,with main cavity beam loading

It is worthwhile to examine the effect of main cavitybeam loading when there is only a single gap in the fillpattern, even though this is not directly relevant to theALS-U design. With 284 bunches, a gap of 44 buckets, andHHCþMC we get the result of Fig. 23 for chargedensities, to be compared with the case of HHC alone inFig. 24. This result could be obtained with the full currentof 500 mA. The graphs show 6 bunches at the head ofthe train (right), middle of the train (middle), and end of thetrain (left). The bunch lengthening is smaller and thecentroid displacement greater when the MC is included.The comparison of bunch lengthenings is shown in Fig. 25.

FIG. 19. Fill C2, HHC alone, Iav ¼ 476.2 mA.

FIG. 20. Two bunches in fill C2 with HHC and its higher ordermode. Iav ¼ 500 mA.

FIG. 21. HHCþ MCþ SR, Iav ¼ 471.9 mA.

FIG. 22. HHCþMC, Iav ¼ 471.9 mA.

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VII. CONCLUSIONS AND OUTLOOK

Continuing the investigation of Ref. [1] we haveextended the physical model to include the effect of themain accelerating cavity in its fundamental mode, pre-viously omitted. We introduced a new algorithm to adjustthe parameters of the rf generator voltage so as tocompensate the voltage induced in the cavity by thebeam, thus putting the net accelerating voltage at a desiredvalue. When the cavity is excited by a bunch train withgaps this compensation implies a modification of thebunch profiles, which is produced automatically in ourscheme.We illustrated the outcome for parameters of the forth-

coming ALS-U storage ring, revisiting examples treated in[1] without the main cavity. The results are similar in modogrosso, but there are significant quantitative differences,especially in cases of overstretching of bunches. Generallyspeaking there is more bunch distortion and less symmet-rical patterns in the bunch trains, and the rms bunchlengthening is a bit smaller and much more variable alongthe train. Correspondingly, the Touschek lifetime increaseis smaller and more variable over a train.We have not tried to model the feedback system that

compensates the beam loading in practice. Our aim wasonly to show the theoretical existence of an equilibriumstate with precise compensation in place.It was disappointing, and somewhat surprising, to find

that the Newton iteration to solve the coupled Haïsinskiequations encounters convergence difficulties at large cur-rent (near the design current) when either the main cavitywake or the short range wake is added to the HHC wake.Our failure to find an equilibrium in some cases of high

current may be due to a failure of technique, not necessarilyan indication that no equilibrium exists. At high current weare trying to achieve convergence of the Newton iterationclose to a singularity of the Jacobian, but not squarely onthe singularity. In this case it is crucial to have a startingguess sufficiently close to a solution, but in practice therequired degree of closeness is unknown. We made someefforts to improve the guess by a seemingly carefulcontinuation in current from the last good solution, butthere was no clear success.

ACKNOWLEDGMENTS

I thank Teresia Olsson for a helpful correspondence, DanWang for her wake potential, and Tianhuan Luo forinformation on the HHC design. Marco Venturini posedthe main cavity compensation problem in general terms.Karl Bane encouraged the study of the short range wake.This work was supported in part by the U.S. Department ofEnergy, Contract No. DE-AC03-76SF00515. My work isaided by an affiliation with Lawrence Berkeley NationalLaboratory as Guest Senior Scientist.

FIG. 25. Single gap, bunch lengthening ratio, for HHCþMC(blue) and with HHC alone (red).

FIG. 23. HHC þMC, single gap, Iav ¼ 500 mA.

FIG. 24. HHC, single gap, Iav ¼ 500 mA.

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APPENDIX: DIAGONAL TERMS IN THE POTENTIAL

Here we find the formula for a generic term in the first sum of (12). For this we revert to the notation used in the case of asingle resonator.

The term in question is the last term of (51) in [1], defined through (60) of that paper, as follows:

Udi ðziÞ ¼

e2NωrRsηξiQ

�Zzi

0

dζZ

ζ

−Σexpð−krðζ − uÞ=2QÞ cosðkrðζ − uÞ þ ψÞρiðuÞdu

þZ

zi

0

dζZ

Σ

ζexpð−krðζ − uþ CÞ=2QÞ cosðkrðζ − uþ CÞ þ ψÞρiðuÞdu

: ðA1Þ

The repeated integrals can be replaced by single integrals through integration by parts. First apply the double angle formulato the cosine, so as to bring out factors of cosðkruÞ and sinðkruÞ. The u-integrals involving those factors are functions of ζ,which are to be differentiated in the partial integration with respect to ζ. The corresponding integration with respect to ζ isdone with the help of (55) and (56) (as indefinite integrals) in [1]. The result is

Udi ðziÞ ¼

ce2NRsηξiQð1þ ð1=2QÞ2Þ ½I1 þ I2�;

I1 ¼Z

zi

−Σexpðkru=2QÞ½aðziÞ cosðkruÞ þ bðziÞ sinðkruÞ�ρiðuÞdu −

�sinψ −

1

2Qcosψ

�Zzi

−ΣρiðuÞdu;

I2 ¼Z

Σ

zi

expðkru=2QÞ½aðzi þ CÞ cosðkruÞ þ bðzi þ CÞ sinðkruÞ�ρiðuÞdu

þ expð−krC=2QÞ�sinðkrCþ ψÞ − 1

2QcosðkrCþ ψÞ

Zzi

−ΣρiðuÞdu;

aðzÞ ¼ expð−krz=2QÞ�sinðkrzþ ψÞ − 1

2Qcosðkrzþ ψÞ

;

bðzÞ ¼ − expð−krz=2QÞ�cosðkrzþ ψÞ þ 1

2Qsinðkrzþ ψÞ

: ðA2Þ

Here we have dropped and added terms independent of zi,which only affect the normalization (19), and have used thedouble angle formula in reverse to consolidate some terms.Writing

RΣzi¼ R

Σ−Σ −

R zi−Σ, we see that there are three differ-

ent integrals to evaluate,

Zzi

−Σ½1; expðkru=2QÞ cosðkruÞ;

expðkru=2QÞ sinðkruÞ�ρiðuÞdu; ðA3Þ

which can be built up stepwise on a mesh in zi. Thus we cancompute and store the diagonal terms on the mesh innegligible time. Note that I2 is totally negligible for thesmall Q that we encounter in representing the geometricwake, owing to the tiny prefactor expð−krC=2QÞ.Summing (A2) over the nr choices of the resonator

parameters kr; Rs; Q; η;ψ we obtain the first termof (12).

[1] R. Warnock and M. Venturini, Equilibrium of an arbitrarybunch train in presence of a passive harmonic cavity:Solution through coupled Haïssinski equations, Phys.Rev. Accel. Beams 23, 064403 (2020).

[2] C. Steier, A. Anders, J. Byrd, K. Chow, R. Duarte, J. Jung,T. Luo, H. Nishimura, T. Oliver, J. Osborn et al., Rþ Dprogress towards a diffraction limited upgrade of the ALS, inProc. IPAC2016, Busan, Korea (JACoW, Geneva, 2016).

[3] C. Steier, A. Allezy, A. Anders, K. Baptiste, J. Byrd,K. Chow, G. Cutler, R. Donahue, R. Duarte, J.-Y. Junget al., Status of the conceptual design of ALS-U, in Proc.IPAC2017, Copenhagen, Denmark (JACoW, Geneva, 2017).

[4] D. Wang, Lawrence Berkeley National Laboratory (privatecommunication). This is from work in progress.

[5] T. Luo, Lawrence Berkeley National Laboratory (privatecommunication).

[6] F. J. Cullinan, Å. Andersson, and P. F. Tavares, Harmonic-cavity stabilization of longitudinal coupled-bunch instabi-litiess with a nonuniform fill, Phys. Rev. Accel. Beams 23,074402 (2020).

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