Equilibrium of an Arbitrary Bunch Train in the Presence of Multiple Resonator Wake Fields
EEquilibrium of an Arbitrary Bunch Train in thePresence of Multiple Resonator Wake Fields
Robert Warnock ∗ SLAC National Accelerator Laboratory,Stanford University, Menlo Park, CA 94025, USA andDepartment of Mathematics and Statistics,University of New Mexico, Albuquerque, NM 87131, USA
Abstract
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 bunchlengthening is complicated by the presence of required gaps in the bunch train. In a recent paperthe author and Venturini studied the effect of various fill patterns by calculating the charge densitiesin the equilibrium state, through coupled Ha¨ıssinski equations. We assumed that the only collectiveforce was from the beam loading (wake field) of the harmonic cavity in its lowest mode. The presentpaper improves the notation and organization of the equations so as to allow an easy inclusion ofmultiple resonator wake fields. This allows one to study the effects of beam loading of the mainaccelerating cavity, higher order modes of the cavities, and short range geometric wakes representedby low- Q resonators. As an example these effects are explored for ALS-U. The compensation ofthe induced voltage in the main cavity, achieved in practice by a feedback system, is modeledby adjustment of the generator voltage through a new iterative scheme. Except in the case of acomplete fill, the compensated main cavity beam loading has a substantial effect on the bunchprofiles and the Touschek lifetimes. A Q = 6 resonator, approximating the effect of a realisticshort range wake, is also consequential for the bunch forms. PACS numbers: ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . acc - ph ] J a n . INTRODUCTION This is a sequel to Ref.[1], in which we explored the action of a higher harmonic cavity(HHC), a standard component of 4th generation synchrotron light sources, employed tolengthen the bunch and reduce the effect of Touschek scattering. In that work we introducedan effective scheme to compute the equilibrium state of charge densities in an arbitrary bunchtrain. The train is allowed to have arbitrary gaps and bunch charges. We chose the simplestpossible physical model, in which the only induced voltage (wake field) is due to the lowestmode of the HHC. We write V r for this voltage, the notation designating “resonator, 3rdharmonic”. We recognized, however, that excitation of the main accelerating cavity (MC)by the bunch train produces an induced voltage V r of comparable magnitude, the effectdescribed as beam loading . Our excuse for omitting V r was that in practice it is largelycancelled by adjusting the rf generator voltage V g through a feedback system. The sum of V r and V g should closely approximate V rf , the desired accelerating voltage.In real machines there are always gaps in the bunch train, and that leads to varying bunchprofiles and centroid displacements along the train. At first sight this would suggest that V r would be different for different bunches, so that compensation could only be partial, perhapsonly manifest in some average sense. On the contrary, we shall calculate an equilibriumstate in which the compensation is essentially perfect for all bunches. This happens byan adjustment of the charge densities of all bunches, to new forms that sometimes differsubstantially from those without the MC.
The adjustment is achieved automatically througha new algorithm presented here. This iterative procedure minimizes a mean square deviationof V r + V g from V rf , summed over all bunches, as a function of two generator parameters,which are equivalent to amplitude and phase.It is not clear that this is a faithful model of the feedback mechanism, which could conceiv-ably amount to a weaker constraint on the bunch profiles. Nevertheless, this study clarifiesthe mathematical structure of the problem, and appears to be a worthwhile preliminary toa full time-dependent model of the system including a realistic description of feedback.Beside the main cavity, we also assess the role of the short range wake field from geometricaberrations in the vacuum chamber, and higher order modes in the HHC. These effects areadded with the help of improvements in notation and organization of the equations.Extensive numerical results are reported with parameters for ALS-U. For consistency with2he previous work the parameters chosen are partially out of date as the machine designstands at present, but that will not greatly affect our pricipal conclusions. Although thequalitative picture of the model with HHC alone is still in place, there are large quantitativechanges. Even then, we underestimate the full effects, because we can only get convergenceof our iterative method when the current is a few percent less than the design current.In Section II we briefly recall our previous algorithm for solving the coupled Ha¨ıssinskiequations. Section III introduces the improved notation and organization which allows aneasy inclusion of multiple resonator wake fields. Section IV enlarges the system of equationsto provide a calculation of the diagonal terms in the potential, thus overcoming a limitationof the previous formulation. Section V describes the method for determining the generatorparameters so as to compensate the induced voltage in the main cavity. Section VI, withseveral subsections, reports numerical results for the case of ALS-U [2, 3], always makingcomparisons to results with only the HHC in place. Subsection VI A treats the case ofa complete fill, illustrating the compensation of the main cavity in the simplest instance.Subsection VI B considers a partial fill with distributed gaps, as proposed for the machine.Subsection VI C is concerned with over-stretching by reduction of the HHC detuning. Sub-section VI D explores the effect of the short range wake field with a realistic wake potential.Subsection VI E checks the effect of the principal higher order mode of the HHC. SubsectionVI F presents our closest approach to a realistic model, including the harmonic cavity, thecompensated main cavity, and the short range wake, altogether. Subsection VI G examinesthe effect of the main cavity when there is only a single gap in the bunch train. Section VIIreviews our conclusions and possibilities for further work. Appendix A derives the expressionfor the diagonal terms in the potential, for resonators of arbitrary Q . II. SUMMARY OF METHOD TO COMPUTE THE EQUILIBRIUM CHARGEDENSITIES
In [1] we derived a set of equations to determine the equilibrium charge densities of n b bunches, which may be stated succinctly as follows: F ( ˆ ρ, I ) = 0 . (1)3ere I is the average current and ˆ ρ is a vector with 2 n b real components, consisting of thereal and imaginary parts of ˆ ρ i ( k r ), where k r is the wave number of the lowest resonantmode of the 3rd harmonic cavity. These quantities are defined in terms of the beam framecharge densities ρ i ( z ), normalized to 1 on its region of support [ − Σ , Σ], asˆ ρ i ( k ) = 12 π (cid:90) Σ − Σ exp (cid:0) − ikz (1 + i/ Q ) (cid:1) ρ i ( z ) dz , (2)where Q is the quality factor of the cavity. The vector in (1) is arranged as follows:ˆ ρ = (cid:2) Re ˆ ρ ( k r ) , · · · , Re ˆ ρ n b ( k r ) , Im ˆ ρ ( k r ) , · · · , Im ˆ ρ n b ( k r ) (cid:3) . (3)Accordingly, F in (1) is a real vector with 2 n b components, so that we have 2 n b nonlinearalgebraic equations in 2 n b unknowns, 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 persistently written all equations for general Q for later applications involvinglow- Q resonators.In (1) the diagonal terms of the induced voltage have been dropped, i.e. the effects on abunch of its own excitation of the cavity. This omission is justified for the typical high Q of an HHC. Our method to handle the diagonal terms in the general case is introduced inSection IV.A solution ˆ ρ of (1) determines the charge densities by the formula of Eq.(50) in [1], ρ i ( z i ) = 1 A i exp (cid:2) − µU i ( z i ) (cid:3) , (4)where U i is the potential felt by the i -th bunch, defined in Eq.(51) of [1]. Here µ and A i are constants, and z i is the beam frame longitudinal coordinate of the i -th bunch. Thepotential U i depends on all components of ˆ ρ , on the mean energy loss per turn U , and onthe parameters of the applied voltage V rf which we write as V rf ( z ) = V sin( k z + φ ) = V (cid:0) cos φ sin( k z ) + sin φ cos( k z ) (cid:1) . (5)We solve (1) by the matrix version of Newton’s iteration, defined in (67) of [1]. We beginat small current I , taking all components of the first guess for ˆ ρ to be the transform (2) of aGaussian with the natural bunch length. We then continue step-wise to the desired current,making a linear extrapolation in current to provide a starting guess for the next Newton4teration at incremented current. The extrapolation is accomplished by solving for ∂ ˆ ρ/∂I from the I -derivative of (1): ∂F∂ ˆ ρ ∂ ˆ ρ∂I + ∂F∂I = 0 . (6) III. FORMALISM FOR MULTIPLE RESONATORS
The scheme allows the inclusion of any number of resonator wake fields, but to do thatconveniently requires some care in notation and organization of the equations. With n r resonators there are 2 n b n r = n u unknowns, which we assemble in one long vector ˜ ρ :˜ ρ = (cid:2) ˜ ρ ( k ) , k = 1 , · · · , n u (cid:3) = (cid:2) Re ˆ ρ ( k r ) , · · · , Re ˆ ρ n b ( k r ) , Im ˆ ρ ( k r ) , · · · , Im ˆ ρ ( k r ) , · · · , Re ˆ ρ ( k r,n r ) , · · · , Re ˆ ρ n b ( k r,n r ) , Im ˆ ρ ( k r,n r ) , · · · , Im ˆ ρ n b ( k r,n r ) (cid:3) . (7)Here k r,n is the resonant wave number of the n -th resonator, and the subscript of ˆ ρ denotesas usual the bunch number.To identify the bunch number and the resonator number for the k -th component of thevector, we define two index maps: ι ( k ) which gives the bunch number and r ( k ) which givesthe resonator number. Namely, ι ( k ) = mod ( k, n b ) if mod ( k, n b ) (cid:54) = 0 n b if mod ( k, n b ) = 0 (8) r ( k ) = (cid:24) k n b (cid:25) . (9)Here (cid:100) x (cid:101) , the ceiling of x , is the least integer greater than or equal to x . We also needtwo projection operators: P re ( k ) which is equal to 1 if k corresponds to a Re ˆ ρ and is zerootherwise, and P im ( k ) which is equal to 1 if k corresponds to a Im ˆ ρ and is zero otherwise.These are expressed in terms of the ceiling of k/n b as follows: P re ( k ) = 12 (cid:20) − ( − (cid:100) k/n b (cid:101) (cid:21) ,P im ( k ) = 12 (cid:20) − (cid:100) k/n b (cid:101) (cid:21) . (10)The potential U j ( z ) for bunch j , generalizing Eq.(51) of [1] to allow n r resonators, is5tated as U j ( z ) = eV k (cid:2) x cos( k z ) − x sin( k z ) − x (cid:3) + U z (11)+ n r (cid:88) n =1 U djn ( z ) + n u (cid:88) k =1 M ( z ) j,k ˜ ρ k , j = 1 , · · · , n b , − Σ ≤ z ≤ Σ . (12)The first term in (11) is e times the integral of the applied voltage, now called the generatorvoltage and written as V g ( z ) = V (cid:2) x sin( k z ) + x cos( k z ) (cid:3) . (13)At x = cos φ , x = sin φ this reduces to the desired V rf of (5). In an amplitude-phaserepresentation we have V g ( z ) = ˜ V sin( k z + ˜ φ ) , ˜ V = ( x + x ) / V , ˜ φ = tan − ( x /x ) . (14)The first term in (12) represents the diagonal contributions, the effect on bunch j of itsown excitation of the resonators, as opposed to excitation by the other bunches which isdescribed by the second term. By writing the latter as a simple matrix-vector product wegreatly simplify the calculation of the Jacobian of the system, making it formally the samefor any number of resonators.Referring to Eqs.(28), (51), (55), (56), (57), (58) of [1], we can write down the matrixelements M ( z ) i,k in the second term of (12). For this we introduce a notation appropriatefor labeling by the index k of (7). Functions of k , defined via the index maps, are labeledwith a tilde: ˜ k r,k = k r,r ( k ) , ˜ ξ k = ξ ι ( k ) , ˜ A k = A ι ( k ) ˜ R sk = R s,r ( k ) , ˜ Q k = Q r ( k ) , ˜ η k = η r ( k ) , ˜ ψ k = ψ r ( k ) , ˜ φ j,k = ˜ k r,k (cid:2) ( m ι ( k ) − m j ) λ + θ j − ,ι ( k ) C (cid:3) ,σ j,k ( z ) = S (cid:0) ˜ k r,k z, ˜ Q k , ˜ φ j,k + ˜ ψ k (cid:1) ,γ j,k ( z ) = C (cid:0) ˜ k r,k z, ˜ Q k , ˜ φ j,k + ˜ ψ k (cid:1) , (15)6here S ( k r z, Q, φ ) = 11 + (1 / Q ) (cid:20) exp( − k r z/ Q ) (cid:18) sin( k r z + φ ) − Q cos( k r z + φ ) (cid:19)(cid:21) z , C ( k r z, Q, φ ) = 11 + (1 / Q ) (cid:20) exp( − k r z/ Q ) (cid:18) cos( k r z + φ ) + 12 Q sin( k r z + φ ) (cid:19)(cid:21) z . (16)The result for the matrix from (51) and (57) of [1] is seen to be (noting that ω r /k r = c ) M ( z ) j,k = 2 πce N ˜ η j ˜ R sj ˜ Q j (1 − δ j,ι ( k ) ) ˜ ξ k exp( − ˜ φ j,k / Q k ) (cid:2) P re ( k ) σ j,k ( z ) + P im ( k ) γ j,k ( z ) (cid:3) . (17)In the present notation the system of coupled Ha¨ıssinski equations, generalizing (66) of[1], takes the form F j ( ˜ ρ ) = ˜ A j ˜ ρ j − π (cid:90) Σ − Σ (cid:2) P re ( k ) cos(˜ k r,j ζ ) − P im ( k ) sin(˜ k r,j ζ ) (cid:3) · exp (cid:2) ˜ k r,j ζ/ Q j − µ U ι ( j ) ( ζ ) (cid:3) dζ = 0 , j = 1 , · · · , n u . (18)The normalization integral appearing in the first term is˜ A j = (cid:90) Σ − Σ exp (cid:2) − µ U ι ( j ) ( ζ ) (cid:3) dζ . (19)We require the Jacobian matrix [ ∂F j /∂ ˜ ρ k ] for the solution of (18) by Newton’s method,assuming that the diagonal terms are fixed. This is found immediately from (12), (18), and(19) as ∂F j ∂ ˜ ρ k = ˜ A j δ j,k − µ (cid:90) Σ − Σ exp (cid:2) − µU ι ( j ) ( ζ ) (cid:3) M ( ζ ) j,k · (cid:20) ˜ ρ j − π (cid:2) P re ( k ) cos(˜ k r,j ζ ) − P im ( k ) sin(˜ k r,j ζ ) (cid:3) exp (cid:2) ˜ k r,j ζ/ Q j (cid:3)(cid:21) dζ . (20)The compact expressions in (17), (18), and (20) are quite convenient for coding, andlead to a short program to solve the Ha¨ıssinski equations with any number of resonators.For ζ at n p mesh points z i used in the integrals we have the array M ( i, j, k ) = M ( z i ) j,k ofmanageable dimension n p × n b × n u which can be computed and stored at the top, outsidethe Newton iteration.For the work of the following section we also need the induced voltage from the maincavity, which we designate as the first resonator in the list ( n = 1). For the j -th bunch this7akes the form V r j ( z ) = − πceN k r R s η Q (cid:20) n b (cid:88) k =1 (1 − δ j,ι ( k ) ) ˜ ξ k exp( − (˜ k r,k z + ˜ φ j,k ) / Q k ) (cid:18) P re ( k ) cos(˜ k r,k z + ˜ φ j,k + ˜ ψ k ) − P im ( k ) sin(˜ k r,k z + ˜ φ i,k + ˜ ψ k ) (cid:19) ˜ ρ k + v d j ( z ) (cid:21) . (21)The diagonal term v d j can be evaluated in terms of integrals derived in Appendix A. IV. THE FULL SYSTEM OF EQUATIONS WITH DIAGONAL TERMS
Through (18) we have a system of n u algebraic equations for determination of ˜ ρ , providedthat the diagonal terms in U i are given. The latter are functionals of the charge densities ρ i ( z i ), from which it follows that (18) can be stated in vector notation as˜ ρ = A ( ˜ ρ, ρ, I ) . (22)On the other hand, the ρ i ( z i ) are determined in turn as solutions of integral equationsprovided that ˜ ρ is given. The integral equations are like normal single-bunch Ha¨ıssinskiequations, but with a background potential determined by ˜ ρ , namely ρ i ( z i ) = 1 A i exp (cid:20) − µU i ( z i , ρ i , ˜ ρ ) (cid:21) , i = 1 , · · · , n b . (23)In vector notation ρ = B ( ρ, ˜ ρ, I ) .. (24)The potential U i depends on the ρ i through its diagonal terms, in the first sum in (12). Ourprocedure will be to interleave the solution of (22) at fixed ˜ ρ , by the usual Newton method,with the solution of (24) at fixed ˜ ρ . If this algorithm converges we shall have consistencybetween ρ and ˜ ρ and a solution of the full system.It turns out, most fortunately, that the solution of (24) is obtained by plain iteration aswould be applied to a contraction mapping, ρ ( n +1) = B ( ρ ( n ) , ˜ ρ, I ) . (25)In our application this usually converges to adequate accuracy in just one step, or three atmost, and takes negligible time. 8his scheme based on (22) and (24) is used in all calculations reported below. It replacesthe method used in [1], which was to evaluate the diagonal terms from the value of ρ fromthe previous Newton iterate. That works only for high- Q resonators, so is not adequate forhandling the short range machine wake. V. ALGORITHM TO ADJUST THE GENERATOR PARAMETERS ( x , x ) We wish to choose ( x , x ) so as to minimize, in some sense, the difference V rf ( z i ) − V g ( z i , x , x ) − V r i ( z i , x , x ) , (26)for all i = 1 , · · · , n b . A reasonable and convenient choice for an objective function tominimize is the sum of the squared L norms of the quantities (26). With a normalizingfactor to make it dimensionless and of convenient magnitude that is f ( x , x ) =12Σ V n b (cid:88) i =1 (cid:90) Σ − Σ (cid:20) V (cos φ − x ) sin( k z ) + V (sin φ − x ) cos( k z ) − V r i ( z, x , x ) (cid:21) dz . (27)The region of integration [ − Σ , Σ] is the same as that used in the definition of the potential U i .Note that the minimum of f cannot be strictly zero, since V r is sinusoidal with wavenumber k r , whereas the other terms are sinusoidal with a slightly different wave number k .Let us adopt the vector notation x = ( x , x ) with norm | x | = | x | + | x | . The equationsto solve now depend on x , having the form F ( ˜ ρ, I, x ) = 0 . (28)To avoid notational clutter we suppress reference to the diagonal terms, leaving it understoodthat a solution of (28) for ˜ ρ actually involves the scheme of the previous session. As usualwe solve for ˜ ρ , for an increasing sequence of I -values. The scheme will be to minimize f ( x ) at each I , thus providing a new x = arg min f to be used at the next value of I . As will nowbe explained, the minimization will also be done iteratively, so that we have an x -iterationembedded in the ˜ ρ -iteration. 9e wish to zero ∇ x F , which is to find x to solve the equations n b (cid:88) i =1 (cid:90) Σ − Σ (cid:20) V (cos φ − x ) sin( k z ) + V (sin φ − x ) cos( k z ) − V r i ( z, x ) (cid:21) × V sin( k z ) + ∂ x V r i ( z, x ) V cos( k z ) + ∂ x V r i ( z, x ) dz = . (29)To solve (29) a first thought might be to apply Newton’s method, starting at some lowcurrent and choosing the zero current solution (cos φ , sin φ ) as the first guess. This wouldbe awkward, however, since it would involve the second derivatives of V r i with respect to( x , x ). The first derivatives must already be done by an expensive numerical differentiation,and the second numerical derivative would be error prone and even more expensive. Instead,let us assume that we have a first guess ( x , x ) and suppose that in a small neighborhood ofthat point the first derivatives of V r i can be regarded as constant. Then second derivativesare zero and the Taylor expansion of V r i gives two linear equations to solve for ( x , x ),namely a a a a x x = b b , (30)where a = (cid:88) i (cid:90) α i ( z, x ) dz , a = (cid:88) i (cid:90) α i ( z, x ) dz ,a = a = (cid:88) i (cid:90) α i ( z, x ) α i ( z, x ) dz ,b = (cid:88) i (cid:90) α i ( z, x ) β i ( z, x ) dz , b = (cid:88) i (cid:90) α i ( z, x ) β i ( z, x ) dz , (31)with α i ( z, x ) = V sin( k z ) + ∂ x V r i ( z, x ) ,α i ( z, x ) = V cos( k z ) + ∂ x V r i ( z, x ) ,β i ( z, x ) = − V r i ( z, x ) + ∇ x V r i ( z, x ) · x + V sin( k z + φ ) (32)(33)10y (30) we have an update x → x which establishes the pattern of the general iterate x ( k ) → x ( k +1) . This will be carried to convergence in the sense | x ( k +1) − x ( k ) | < (cid:15) x , with asuitable (cid:15) x to be determined by experiment. Each iterate requires a value for V r i and for ∇ x V r i , which we compute numerically by a divided difference, ∂V r i ∂x ( z, x ) ≈ V r i ( z, x + ∆ x, x ) − V r i ( z, x , x )∆ x . (34)Thus one x -iteration requires three ˜ ρ -iterations to provide the necessary values of V r i (whichare constructed from ˜ ρ ). The first ˜ ρ iteration to find V r i ( z, x , x ) produces a ˜ ρ which is avery good guess to start the remaining two iterations to make the derivatives, which thenconverge quickly.The choice of ∆ x in (34) requires a compromise between accuracy and avoiding round-offerror. We found that ∆ x = 10 − was widely satisfactory, whereas success with smaller valuesdepended on the circumstances. VI. NUMERICAL RESULTS WITH AND WITHOUT THE MAIN CAVITY
As in [1] we illustrate with parameters for ALS-U [2, 3], the forthcoming AdvancedLight Source Upgrade. Although the machine design is not yet final, one provisional set ofparameters for our main cavity (actually the effect of two cavities together) is as follows: R s = 0 . M Ω , Q = 3486 , δf = f r − f = − . kHz (35)Here the shunt impedance R s and quality factor Q are loaded values, the unloaded valuesdivided by 1 + β , with coupling parameter β = 7 . U =217keV, even though a value of 330 keV may be contemplated for the set (35). A. Complete Fill
We first take the case of a complete fill, thus n b = h = 328. The average current is to be500 mA, which we reach in 8 steps starting from 200 mA. The CPU time is 15 minutes, ratherthan 20 seconds for the calculation without the main cavity. The increase is mostly due toa much slower convergence of the ˜ ρ -iteration, the x -iteration being a minor factor in CPU11ime. To save time we gave (cid:15) x the rather large value of 0.05, but then made a refinement to (cid:15) x = 10 − at the final current, in an extra 2 minute. The steepness of the objective function f ( x , x
2) of (27) is extraordinary, having values around 10 in the sequence with (cid:15) x = 0 . FIG. 1: Charge density for complete fill at 500 mA, with compensated main cavity (blue) andwithout main cavity (red).
In Fig.2 we show the compensation mechanism. The sum of the generator voltage V g andthe induced voltage V r from the main cavity is the orange curve. The latter deviates fromthe desired effective voltage V rf by less than 2%, as is seen Fig.3.The phasor of the generator voltage moves closer to π/ (cid:112) x + x increases from 1 to 1.0245, in comparison to the phasor of V rf .The corresponding values of ( x , x ) are( x , x ) = (cos φ , sin φ ) = ( − . , . → ( x , x ) = ( − . , . . (36)12 IG. 2: MC induced voltage V r , generator voltage V g and their sum.FIG. 3: Relative deviation of V r + V g from V rf . B. Partial fill C2 with distributed gaps
Next we take a partial fill with distributed gaps, labeled as fill C2; see Section XIII-Cof [1]. There are 284 bunches in 11 trains, with 4 empty buckets between trains. Thereare 9 trains of 26 and 2 of 25, with the latter positioned at opposite sides of the ring. Allbunches have the same charge. As in the preceding example we start the calculation at lowaverage current and advance in steps trying to reach the desired 500 mA. The convergence13f iterations is at first similar to that of the preceding case, but begins to falter around 430mA average current, at which point the convergence of the ˜ ρ -iteration becomes problematic.By taking smaller and smaller steps in current we can reach 496 mA, but beyond that pointthe Jacobian matrix of the system appears to approach a singularity, as is indicated by itsestimated condition number having a precipitous increase, from 700 at the last good solutionto 2900 at a slightly higher current. Nevertheless, the x -iterations continue to converge aslong as the ˜ ρ -iterations do. In the following, graphs are plotted for the maximum achievablecurrent, stated in figure captions. Now the plots of V g and V r and their sum look exactly the same as in Fig.2,for every bunch. The minimization of f ( x , x ) has caused the bunch forms torearrange themselves so that the compensation is essentially perfect for everybunch. The deviation of V r + V g from V rf , scarcely visible on the scale of Fig.2,varies from bunch to bunch, but is still less than 3% for all bunches. Fig.4 shows 9 bunch profiles in one train, to be compared with the corresponding resultswithout the main cavity in Fig. 5. The main cavity causes considerably more bunch dis-tortion along the train, and also a bigger variation in the rms bunch lengths, as is seen inFig.6. The plots show the ratio of bunch length to the natural bunch length. The head ofthe train is on the right, with the highest bunch number.The corresponding results for the bunch centroids is seen in Figs.7. Again the deviationfrom the case without the main cavity is quite substantial.The main point of practical interest is the increase in Touschek lifetime achieved throughthe bunch stretching caused by the HHC. Again, the MC has a sizeable effect in reducingthe lifetime and in causing a larger variation along a train. This is shown in Fig.8 whichgives the ratio of the lifetime τ to the lifetime τ without the MC.We next consider the same fill pattern with 11 trains, but with a taper in the bunchcharges putting more charge at the ends, according to a power law as shown in Fig. 15 of[1]. This is an example of invoking guard bunches to reduce the effect of gaps. As is seenin Figs.9 and 10, the guarded inner bunches, which resemble that of the complete fill, arelittle affected by the MC. The strong asymmetry between the front and back of the trainis perhaps surprising, but it should be noticed that Fig.10 already shows an appreciablefront-back asymmetry. The strong amplification of this asymmetry by the MC is in linewith its big effects seen generally. 14 IG. 4: Charge densities in a train of 26,surrounded by gaps of 4 buckets, fill C2,MC beam loading included, I av = 496 mA. FIG. 5: Charge densities in a train of 26,surrounded by gaps of 4 buckets, fill C2,MC beam loading omitted, I av = 496 mA.FIG. 6: Bunch length increases in a train of 26, surrounded by gaps of 4 buckets, with main cavitybeam loading (blue) and without (red). I av = 496 mA. The plot is the ratio of bunch length σ tothe natural bunch length σ . C. Decrease of HHC detuning for over-stretching
There is practical interest in the possibility of over-stretching for an additional increase inthe Touschek lifetime. This entails a decrease in the detuning of the HHC, which producesa larger r.m.s. bunch length but a bunch profile with a dip in the middle, thus a double15
IG. 7: Centroids < z > in a train of 26, surrounded by gaps of 4 buckets, with cavity beamloading (blue) and without (red). I av = 496 mAFIG. 8: Touschek lifetime increase along a train, with compensated MC (blue) and without (red). I av = 496 mA. peak. In our case a decrease from df = 250 . df = 235 kHz produces a doublepeak in the model without the MC at full current, as is seen in Fig.4 of [1]. We would liketo know how this setup looks with the compensated main cavity in play. Not surprisingly,the convergence of our iterative solution breaks down at a lower current than in the case ofthe normal detuning; the stronger the bunch distortions the poorer the convergence. With16 IG. 9: Case of tapered bunch charges,MC beam loading included, I av = 496 mA. FIG. 10: Case of tapered bunch charges,MC beam loading omitted, I av = 496 mA. df = 235 kHz and the MC we can only reach 474.5 mA, which is not enough to see adouble peak. Nevertheless it is useful to compare the result at that current with the resultin absence of the MC, as displayed for 9 bunches in a train of 27 in Figs.11 and 12. FIG. 11: Fill C2 with HHC + MC,detuning df = 235 kHz, I av = 474 . df = 235 kHz, I av = 474 . Even at a current significantly less that the 500 mA design current the distortion due tothe main cavity is quite large, which leads to the conclusion that the main cavity mustbe included in a realistic simulation of over-stretching. . Effect of the short range wake field The short range wake field from various unavoidable corrugations in the vacuum chamberretains importance in the latest storage rings, in spite of the best efforts to reduce it. Sinceit can cause substantial bunch distortion in the absence of an HHC, we would like to knowhow much it affects the operation of the HHC. A result for the longitudinal wake potential atALS-U, from a detailed computation by Dan Wang [4], is shown in Fig.13. The corresponding
FIG. 13: Wake potential (pseudo - Green function) for the ALS-U storage ring, computed with a1 mm driving bunch. impedance, Z ( f ) = 1 c (cid:90) ∞−∞ e − ikz W ( z ) dz , f = kc/ π , (37)is plotted in Fig.14.In Ref.[1] we suggested that a low- Q resonator wake could be treated on the same footingas the high- Q resonators, and for that reason we wrote all equations for a general value of Q .We recognized, however, that the diagonal term in the potential would now be dominant,while being nearly negligible in the high- Q case. It could not be treated by the method usedin [1], but is easily handled by the presently adopted method of Section IV.For the equilibrium state, the impedance at f >
20 GHz is irrelevant, even though itcould have a role out of equilibrium. This assertion follows from the fact that the frequencyspectrum of our calculated charge densities never extends beyond 15 GHZ, no matter which18
IG. 14: Longitudinal impedance Z ( f ) for ALS-U. wake fields are included. Consequently, a reasonable step is to concentrate on the first bigpeak at 11.5 GHz. The wake potential in our equations (defined in (19) of [1]) is based onan impedance as follows, which is of Lorentzian form with half-width Γ / Z ( f ) = iR s Γ2 (cid:20) f − f r + i Γ / f + f r + i Γ / (cid:21) = Z ( − f ) ∗ , Γ / f r / Q . (38)Figures 15 and 16 show a fit to (38) with parameters as follows: f r = 11 .
549 GHz , R s = 5730 Ω , Q = 6 . (39)The fit is rough in the imaginary part, but probably good enough to estimate the magnitudeof the effect of the short range wake.Discussions of low- Q resonator models in the literature usually invoke the impedance ofan LRC circuit, Z ( f ) = R/ (1 + iQ ( f r /f − f /f r )), often with Q near 1. As is illustrated inFigures 15 and 16, in our case with Q = 6 the LRC model does not give a better fit thanthe simpler Lorentzian, except for enforcing Z (0) = 0. At the expense of some complicationour equations could be modified to accommodate the LRC form, but that appears to beunnecessary, at least in the present example.Henceforth, the impedance from (38) and (39) will be referred to as SR (short range).Taking first a complete fill, and including just the HHC and SR, we get the result of Fig.17.19 IG. 15: Fit of Re Z to Lorentzian andLRC circuit formulas. FIG. 16: Fit of Im Z to Lorentzian andLRC circuit formulas.FIG. 17: Charge density for a complete fill, with HHC plus the first peak in the short rangeimpedance (blue), and with HHC alone (red). I av = 500 mA. Next we consider the partial fill C2 with distributed gaps as treated in the previoussection. Figures 18 and 19 show the results for HHC+SR and HHC alone. As expected,the effects of SR are more pronounced in the partial fill than in the complete fill. Corre-spondingly, the maximum current achieved is 472.6 mA. As in previous cases we expect asubstantially larger effect at the design current of 500 mA.20
IG. 18: Fill C2, HHC + SR, I av = 476 . I av = 476 . E. Higher order mode (HOM) of the harmonic cavity
At the present stage of design the most prominent longitudinal HOM of the HHC forALS-U is a TM011 mode with the following parameters [5]: R s = 3000 Ω , Q = 80 , f r = 2 .
29 GHz (40)A calculation for fill C2 with the HHC and this HOM gave the result of Fig.20. The effectof the HOM on the charge densities 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 in longitudinal coupled-bunch instabilities is discussed in Ref.[6].
F. The full model: HHC+MC+SR.
We are now prepared to include the harmonic cavity, the compensated main cavity, andthe short range wake, altogether. The convergence of the Newton sequence suffers even morethan in the previous cases, and the continuation in current reaches only I av = 471 . IG. 20: Two bunches in fill C2 with HHC and its higher order mode. I av = 500 mA.FIG. 21: HHC+MC+SR, I av = 471 . I av = 471 . G. The case of a single gap, with main cavity beam loading
It is worthwhile to examine the effect of main cavity beam loading when there is only asingle gap in the fill pattern, even though this is not directly relevant to the ALS-U design.With 284 bunches, a gap of 44 buckets, and HHC+MC we get the result of Fig.23 forcharge densities, to be compared with the case of HHC alone in Fig.24. This result could beobtained with the full current of 500 mA. The graphs show 6 bunches at the head of the train(right), middle of the train (middle), and end of the train (left). The bunch lengthening issmaller and the centroid displacement greater when the MC is included. The comparison of22
IG. 23: HHC+MC, single gap, I av = 500 mA. FIG. 24: HHC, single gap, I av = 500 mA. bunch lengthenings is shown in Fig.25. FIG. 25: Single gap, bunch lengthening ratio, for HHC+MC (blue) and with HHC alone (red).
VII. CONCLUSIONS AND OUTLOOK
Continuing the investigation of Ref.[1] we have extended the physical model to includethe effect of the main accelerating cavity in its fundamental mode, previously omitted. Weintroduced a new algorithm to adjust the parameters of the rf generator voltage so as tocompensate the voltage induced in the cavity by the beam, thus putting the net accelerating23oltage at a desired value. When the cavity is excited by a bunch train with gaps thiscompensation implies a modification of the bunch profiles, which is produced automaticallyin our scheme.We illustrated the outcome for parameters of the forthcoming ALS-U storage ring, re-visiting examples treated in [1] without the main cavity. The results are similar in modogrosso , but there are significant quantitative differences, especially in cases of overstretchingof bunches. Generally speaking there is more bunch distortion and less symmetrical patternsin the bunch trains, and the rms bunch lengthening is a bit smaller and much more variablealong the train. Correspondingly, the Touschek lifetime increase is smaller and more variableover a train.We have not tried to model the feedback system that compensates the beam loading inpractice. Our aim was only to show the theoretical existence of an equilibrium state withprecise compensation in place.It was disappointing, and somewhat surprising, to find that the Newton iteration to solvethe coupled Ha¨ısinski equations encounters convergence difficulties at large current (nearthe design current) when either the main cavity wake or the short range wake is added tothe HHC wake.A colleague suggested that the failure of convergence might hint at an instability. Oneshould be cautious about such an idea. The issue here is just the existence of an equilibrium.An equilibrium may or may not be stable under time evolution, so stability is a differentissue.Our failure to find an equilibrium in some cases of high current may be due to a failure oftechnique, not necessarily an indication that no equilibrium exists. At high current we aretrying to achieve convergence of the Newton iteration close to a singularity of the Jacobian,but not squarely on the singularity. In this case it is crucial to have a starting guesssufficiently close to a solution, but in practice the required degree of closeness is unknown.We made some efforts to improve the guess by a seemingly careful continuation in currentfrom the last good solution, but there was no clear success.A likely remedy for the convergence failure is to return to the conventional formulationof the Ha¨ıssinski equations as integral equations for the charge densities, in place of thepresent formulation as algebraic equations for Fourier amplitudes. For a single Ha¨ıssinskiintegral equation discretized on a mesh in z -space, the Newton iterative solution is ultra-24obust, converging at currents far beyond realistic values [7]. It seems likely that similargood behavior will hold for the coupled integral equations. The size of the discretized systemdoes not grow with the number of resonator wakes, in contrast to the present system, andthe full z -space description of the short range wake could be invoked in place of the low- Q resonator model.To make this z -space formulation feasible on modest computer resources we can assumethat all bunch sub-trains are identical, and all separated by identical gaps. For the ALS-Uthis would mean artificially increasing the harmonic number from 328 to 330, and having 11trains of 26 separated by gaps of 4 buckets. Then we have 26 independent charge densities,which can adequately be described by 100 mesh points each. Thus the Jacobian of theNewton iteration is 2600 × z -space system. Also, the formalism for multiple resonator wakeswill still be advantageous in the z -space scheme. VIII. ACKNOWLEDGMENTS
I thank Teresia Olsson for a helpful correspondence, Dan Wang for her wake potential,and Tianhuan Luo for information on the HHC design. Marco Venturini posed the maincavity compensation problem in general terms. Karl Bane encouraged the study of the shortrange wake. This work was supported in part by the U. S. Department of Energy, ContractNos. DE-AC03-76SF00515. My work is aided by an affiliation with Lawrence BerkeleyNational Laboratory as Guest Senior Scientist.25 ppendix A: Diagonal terms in the potential.
Here we find the formula for a generic term in the first sum of (12). For this we revert tothe notation used in the case of a single resonator.The term in question is the last term of (51) in [1], defined through (60) of that paper,as follows: U di ( z i ) = e N ω r R s ηξ i Q (cid:20) (cid:90) z i dζ (cid:90) ζ − Σ exp( − k r ( ζ − u ) / Q ) cos( k r ( ζ − u ) + ψ ) ρ i ( u ) du + (cid:90) z i dζ (cid:90) Σ ζ exp( − k r ( ζ − u + C ) / Q ) cos( k r ( ζ − u + C ) + ψ ) ρ i ( u ) du (cid:21) . (A1)The repeated integrals can be replaced by single integrals through integration by parts.First apply the double angle formula to the cosine, so as to bring out factors of cos( k r u )and sin( k r u ). The u -integrals involving those factors are functions of ζ , which are to bedifferentiated in the partial integration with respect to ζ . The corresponding integrationwith respect to ζ is done with the help of (55) and (56) (as indefinite integrals) in [1]. Theresult is U di ( z i ) = ce N R s ηξ i Q (1 + (1 / Q ) ) (cid:2) I + I (cid:3) ,I = (cid:90) z i − Σ exp( k r u/ Q ) (cid:20) a ( z i ) cos( k r u ) + b ( z i ) sin( k r u ) (cid:21) ρ i ( u ) du − (cid:0) sin ψ − Q cos ψ (cid:1) (cid:90) z i − Σ ρ i ( u ) du ,I = (cid:90) Σ z i exp( k r u/ Q ) (cid:20) a ( z i + C ) cos( k r u ) + b ( z i + C ) sin( k r u ) (cid:21) ρ i ( u ) du + exp( − k r C/ Q ) (cid:0) sin( k r C + ψ ) − Q cos( k r C + ψ ) (cid:1) (cid:90) z i − Σ ρ i ( u ) du ,a ( z ) = exp (cid:0) − k r z/ Q (cid:1)(cid:0) sin( k r z + ψ ) − Q cos( k r z + ψ ) (cid:1) ,b ( z ) = − exp (cid:0) − k r z/ Q (cid:1)(cid:0) cos( k r z + ψ ) + 12 Q sin( k r z + ψ ) (cid:1) . (A2)Here we have dropped and added terms independent of z i , which only affect the normal-ization (19), and have used the double angle formula in reverse to consolidate some terms.Writing (cid:82) Σ z i = (cid:82) Σ − Σ − (cid:82) z i − Σ , we see that there are three different integrals to evaluate, (cid:90) z i − Σ (cid:2) , exp( k r u/ Q ) cos( k r u ) , exp( k r u/ Q ) sin( k r u ) (cid:3) ρ i ( u ) du , (A3)26hich can be built up stepwise on a mesh in z i . Thus we can compute and store the diagonalterms on the mesh in negligible time. Note that I is totally negligible for the small Q thatwe encounter in representing the geometric wake, owing to the tiny prefactor exp( − k r C/ Q ).Summing (A2) over the n r choices of the resonator parameters k r , R s , Q, η, ψ we obtainthe first term of (12). [1] R. Warnock and M. Venturini, Equilibrium of an arbitrary bunch train in presence of a passiveharmonic cavity: Solution through coupled Ha¨ıssinski 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. , “R+D progress towards a diffraction limited upgrade of the ALS”, Proc.IPAC2016, Busan, Kroea.[3] C. Steier, A. All´ezy, A. Anders, K. Baptiste, J. Byrd, K. Chow, G. Cutler, R. Donahue,R. Duarte, J.-Y. Jung et al. , “Status of the conceptual design of ALS-U”, Proc. IPAC2017,Copenhagen, Denmark.[4] Dan Wang, Lawrence Berkeley National Laboratory, private communication. This is from workin progress.[5] Tianhuan Luo, Lawrence Berkeley National Laboratory, private communication.[6] F. J. Cullinan, ˚A. Andersson, and P. F. Tavares, Harmonic-cavity stabilization of longitudinalcoupled-bunch instabilitiess with a nonuniform fill, Phys. Rev. Accel. Beams , 074402 (2020).[7] R. Warnock and K. Bane, Numerical Solution of the Ha¨ıssinski Equation for the EquilibriumState of a Stored Electron Beam, Phys. Rev. Accel. Beams , 124401 (2018).[8] R. Warnock, Study of Bunch Instabilities by the Nonlinear Vlasov-Fokker-Planck Equation,Nuc. Instrum. Methods Phys. Res. A , 186 (2006)., 186 (2006).