Cumulative beam breakup in linear accelerators with time-dependent parameters

A formalism presented in a previous paper for the analysis of cumulative beam breakup (BBU) with arbitrary time dependence of the beam current and with misalignment of the cavities and focusing elements [J. R. Delayen, Phys. Rev. ST Accel. Beams 6, 084402 (2003)] is extended to include time dependence of the focusing and coupling between the beam and the dipole modes. Such time dependence, which could result from an energy chirp imposed on the beam or from rf focusing, is known to be effective in reducing BBU-induced instabilities and emittance growth. The analytical results are presented and applied to practical accelerator configurations and compared to numerical simulations.


I. INTRODUCTION
The cumulative beam breakup instability (BBU) in linear accelerators results when a beam is injected into an accelerator with a lateral offset or an angular divergence, and couples to the dipole modes of the accelerating structures [1]. The dipole modes that are excited in a cavity by the previous bunches can further deflect the following bunches and thereby increase the excitation of the dipole modes in the downstream cavities. In this process the transverse displacement can be amplified and lead to a degradation of beam quality and possibly beam loss. This instability is cumulative since the transverse deflection of a particular bunch or particle results from the additive contributions from all the previous bunches.
Cumulative BBU has been studied in the past mostly in the context of high energy electron accelerators where the beam current profiles were comprised of periodic trains of point-like bunches [2][3][4][5][6][7][8] or for high-current quasi-dc beams [9][10][11][12][13]. Growing interest in highcurrent superconducting ion accelerators for spallation sources, where the bunches have a finite length, motivated an investigation of cumulative BBU in linear accelerators with periodic beam current profile [14].
More recently a general analysis of BBU with arbitrary time dependence of the beam current and injection offsets, as well as random displacement of cavities and focusing elements was developed [15,16]. This analysis is extended here to include time dependence of the focusing and of the coupling between the beam and the dipole mode in order to control or suppress the BBU instability and the resulting emittance growth. This can be accomplished either with rf focusing or by introducing an energy spread within a single bunch as originally proposed [17], or, more recently, by introducing an energy spread between the bunches [18].
Several analysis of what is often referred to as BNS damping have been made before [19][20][21][22][23][24][25][26][27][28] usually under simplifying assumptions for the current profile, wake function, or time dependence of the parameters. We present here an analysis that provides an exact solution for arbitrary beam current profile, wakefunction, offset parameters, misalignment of cavities and focusing elements, and time dependence of focusing and BBU coupling strengths. The analytical results are compared to numerical simulations and are found to be in complete agreement.

II. FORMULATION AND GENERAL SOLUTION
In a continuum approximation, the transverse motion of a relativistic beam in a misaligned accelerator under the combined influence of focusing and coupling to the dipole modes can be modeled by In this expression σ = s/L is the distance from the entrance of the accelerator normalized to the accelerator length L; κ is the normalized focusing wavenumber; ζ = ω t − ds/c is the time made dimensionless by an angular frequency ω and measured after the arrival of the head of the beam at location σ; F (ζ) = I(ζ)/I, the current form factor, is the instantaneous current divided by the average current; ω(ζ) is the wakefunction of the dipole modes; ε is the coupling strength between the beam and the dipole modes; d f (σ) and d c (σ) are the lateral displacements of the focusing elements and the cavities, respectively, as a function of location along the accelerator.
The dimensionless BBU coupling strength ε is given by where w 0 is the wake amplitude. With these definitions the wake function w(ζ) is a dimensionless function of a dimensionless variable and includes only the functional dependence on ζ.
The continuum model assumed in Eq. (1) relies on a number of approximations that are addressed in [15]. Equation (1) also assumes a coasting beam in a uniform accelerator but, as shown in Appendix A of [15], an accelerated beam can, under general assumptions, be reduced to a coasting beam with the introduction of appropriate variable and coordinate transformations.

A. Time-independent parameters
When κ and ε are constant (independent of ζ) Eq. (1) can be solved through the use of the Laplace transform with respect to σ: and the solutions for x † (p, ζ) and x(σ, ζ) are [15] x and The functions of f n (ζ), g n (ζ), and h n (ζ) are defined by identical recursion relations with where x 0 (ζ) and x 0 (ζ) are the lateral displacement and angular divergence, respectively, of the beam at the entrance of the accelerator. The normalizing constants x 0 and x 0 are introduced to make the functions h 0 (ζ) and g 0 (ζ) dimensionless.
The functions i n (κ, σ) end j n (κ, σ) are defined in terms of Bessel functions of order integer plus one half and is the convolution of i n (κ, σ) and d(σ).
Applications of these results were presented in [15,16].

B. Time-dependent parameters
When the strengths of the focusing and of the coupling between the beam and the dipole modes are time-dependent [ε(ζ) and κ (ζ)] the beam displacement governed by Eq. (1) is not given by Eq. (5) any more and the procedure for solving Eqs. (1) and (3) needs to be modified. This can be done by splitting the focusing strength κ(ζ) in two parts, one constant and one time-dependent, such that As shown in the Appendix, the displacement x(σ, ζ) and its Laplace transform x † (p, ζ) are then given by where the functions f n (ζ), g n (ζ), h n (ζ), and k n (ζ) satisfy identical recursion relations and Equation (12) gives the transverse displacement, at location σ and time ζ, of a beam of arbitrary current F (ζ), entering the accelerator with lateral offset x 0 (ζ) and angular divergence x 0 (ζ), experiencing transverse forces due to coupling ε(ζ) to cavity dipole modes of wakefield w(ζ) and to focusing κ(ζ), and with displacement along the accelerator d c (σ) of the cavities and d f (σ) of the focusing elements.
There is some arbitrariness in the way the focusing strength κ(ζ) is split in two parts according to Eq. (10). For example, it could be assumed that κ(ζ) has no constant term (κ 0 = 0) and only a time-dependent term. In this case x † (p, ζ) and x(σ, ζ) would be given by where the functions and The arbitrary constantκ is introduced only to make the function k • 0 (ζ) dimensionlesssimilar to f • 0 (ζ), g • 0 (ζ), and h • 0 (ζ) -and the final result given by Eq. (17) is independent of its choice.
While the expressions (11)- (15) and (16) They differ significantly, however, in their rate of convergence, with expressions (16)- (19) converging very slowly. For expressions (11)-(15) to be of practical use, the separation of κ 2 (ζ) in two parts, as given by Eq. (10), needs to be done in such a way that the timedependent part ∆κ(ζ) is kept as small as possible. Since, in practical applications, the focusing strength will change only by a small amount, natural choices for κ 2 0 would be the value of κ 2 (ζ) either at the beginning, the end, or its average value while the beam is in the accelerator.
Equations (11)-(15) are very general. For the remainder of this paper, in order to simplify the equations, we will assume that the accelerator is perfectly aligned [d f (σ) = d c (σ) = 0], that the injection offsets are time independent [x 0 (ζ) = x 0 , x 0 (ζ) = x 0 ], and that the beam was turned on at ζ = 0 [F (ζ < 0) = 0]. With these assumptions x † (p, ζ) and x(σ, ζ) are given by From Eqs. (22) we see that Thus, by choosing we have and As a result the beam progresses in the accelerator as though it was only experiencing a constant focusing force without interaction with the deflecting mode. Equation (24) is the general condition for complete suppression of BBU by imposing a time dependence on the focusing or the coupling strength. This condition is often referred to as autophasing [23].
It should be noted that the autophasing condition Eq. (24) that eliminated the BBU instability was obtained in the case of a perfectly aligned accelerator with time-independent injection offsets. As can be seen from Eq. (12), the autophasing condition would still be ef-

III. SINGLE VERY SHORT BUNCH
The results of the previous section will be first applied to the case of a single, very short bunch. In this case the wakefunction is assumed to be linear [w(ζ) = ζ] and the current density along the bunch is assumed to be constant [F (ζ) = 1]. We will further assume that the BBU coupling strength is constant.
If the time dependence of the focusing is assumed to be quadratic where ζ b is the bunch length, then the functions f * n (ζ) can be easily calculated This, together with Eq. (21), defines completely the displacement x(σ, ζ).
If the focusing modulation parameter η is chosen such that η = ε/(2 κ 2 0 ), then f n>0 (ζ) = 0 and the coupling between the beam and the dipole mode is completely suppressed.
In the case of a linear time dependence of the focusing, the functions f n (ζ) can be obtained through the recursion relations a n,k = ε a n−1,k−2 k(k − 1) − κ 2 0 η a n−1,k−1 .
In particular (31c) This is illustrated in Fig. 1 which shows the shape of a bunch at the exit (σ = 1) of an accelerator for quadratic (left) and linear (right) time dependence of the focusing modulation.
The parameter η represents the amount of modulation from the front (ζ/ζ b = 0) to the tail (ζ/ζ b = 1) of the bunch. The focusing strength is set at κ 0 = 100.5π; this implies that, since σ = 1, the lateral displacement is due entirely to the coupling with the deflecting mode and to the focusing modulation, and not to the betatron motion without focusing modulation.
The strength of the coupling to the dipole mode is assumed to be ε = 0.8. Figure 1 illustrates how the right amount of quadratic modulation can cancel the effect of the coupling to the dipole modes, while a linear modulation can reduce it but not eliminate it.

IV. FINITE TRAIN OF POINT-LIKE BUNCHES
The results of section II will be applied here to a finite train of N identical point-like bunches separated, in the laboratory frame, by τ , so that bunch M is defined by ζ = Mωτ .
The displacement of bunch M is then given by As an example, the analytical results expressed by Eqs. (32) and (33) will be applied to a beam representative of a linear collider. For comparison we will use the same parameters as those used in [15,16,19,20] and which are listed in Table I. Since this is an accelerated beam, the transformations described in Appendix A of [15] will be used. Converting the parameters in Table I to those used in this paper we have ε(0) = w 0 q e L 2 γ(0) mc 2 ω τ = 38.02, κ(0) = 1100 π, ς(σ = 1) = 2/11, and ω τ = 263.014.
This example illustrates the relative effects of a finite Q of the deflecting mode and of a time-dependent focusing strength of the form where M is the bunch number and N is the total number of bunches.
The transverse displacement at location σ for bunch M is given by Eq.   [15,16,19,20] Parameter Value Total initial energy γ(0)mc 2 Table I for the choice of parameters.

APPENDIX: GENERAL SOLUTION OF THE EQUATION OF MOTION
The equation of motion for the transverse displacement is We separate the focusing in two parts, one constant and one time-dependent, such that and we apply the Laplace transform with respect to σ to obtain By analogy with the time-independent case [where κ 2 1 (ζ) = 0] we assume that x † (p, ζ) is of the form where the functions g n (ζ), h n (ζ), k n (ζ), and l n (ζ), are to be determined.
To determine k n (ζ) we assume that x 0 (ζ) = x 0 (ζ) = 0 and d c (σ) = 0, so that x † (p, σ) reduces to and must satisfy (A. 10) This implies that To determine l n (ζ) we assume that x 0 (ζ) = x 0 (ζ) = 0 and d f (σ) = 0, so that x † (p, σ) reduces to x † (p, σ) = −d † c (p) and must satisfy This implies that which leads to If we define the functions f n (ζ) by Since the functions f n (ζ), k n (ζ), and l n (ζ) satisfy the same recursion relation we obtain