Multibunch Motion with Nearest Neighbor Wakefield Coupling

This paper discusses various aspects of multibunch motion in thepresence of nearest neighbor wakefield coupling. Included are the solution to the problem for smooth focusing with equal bunch energies, an explanation for the apparent &mping of the bunch amplitudes that is observed for weak coupling, and a treatment of the problem for discrete focusing based on the moments of the wakefield distributions in the structures. lINTRODUCTION _Many of the designs being considered for the next generation linear collider use multiple bunch operation to achieve the desired luminosity. One limitation in this approach comes from the coupling of the motion of the bunches due to the long-range transverse wakefields generated in the accelerator structures of the linac. If not controlled, these wakefields will produce a large growth in the transverse-motion of the bunches that degrades the luminosity [l-l. Qne means of reducing this growth is by detuning the dipole modes in the cells of the structures so that the sum of the wakefields generated in each structure decohere. At SLAC, for example, the X-band structures being developed for the Next Linear Collider (NLC) will have a 10% Gaussian detuning of the cells to reduce the wakefield sum by two orders of magnitude at the downstream bunch locations [2]. The frequency spread that can be accommodated for such detuning is usually limited, so the decoherence time is generally comparable to the bunch separation time. Hence the nearest downstream .bunch is likely to experience the largest wakefield kick. If damping is used to suppress the dipoles modes, the neighboring bunch is also likely to receive the largest kick. In these cases it may be a good approximation when treating the problem of multibunch motion to consider only the nearest neighbor coupling, that is, the effect of the wakefield generated by each bunch on only its immediate downstream neighbor. In the following sections, we examine some of the characteristics of multibunch motion with this type of coupling. 2 MUTIBUNCH MOTION WITH SMOOTH FOCUSING To formulate a solution to the multibunch motion problem for-ne-arest [email protected] coupling, we first need to define the linac configu&ion ahd the dynamics. To simplify the problem we assume a constant acceleration gradient linac with smooth focusing in which the beta function grows as the square. root of the bunch energy. For specific applications of the results, we will use the linac parameters listed in Table I. For this set, the fractional change in the bunch energy on the distance scale of a betatron wavelength is small enough that the equations of motion can be reduced to a good approximation to those for a zero acceleration gradient linac of length Using the values in the table, this yields a length of 1.9 km compared to the actual linac length of 4.9 km. In this zero acceleration gradient linac, it will be assumed that the bunch energies are constant and equal. Also, each bunch will be treated as a macro-particle of charge It, that has a tmnsverse offset of unity at the beginning of the linac. Table I. NLC Linac Parameters Quantity .. 1 Symbol 1 Value 1 Charge per bunch Initial Beta Function Initial Linac Energy Final Linac Energy Acceleration Gradient Ib PO Eo Ef &I 1.10’0 4m 16 GeV 250 GeV 50 MeVlm To represent the bunch-to-bunch coupling, we treat the wakefield as being independent of position along the linac and characterize its strength by the average of the dipole mode wakefields generated by a bunch trajectory with a fixed transverse offset in a structure. This representation ignores the effect of the variation of the bunch trajectory and wakefields within the structure, but is generally a good approximation when the betatron wavelength is large compared to the structure length (a formalism to include these effects is discussed in section 4). With this assumption, the bunch interaction can be expressed by dei WbbIb ~ xi-1 dzE-3 (2) which relates the angular kick to bunch i per unit length of the linac, d&/dz, to the transverse position, xi-i, of the next upstream bunch. The coupling coefficient is the wakefield strength per unit length at one bunch separation, Wbb, and is normalized to the charge of the leading bunch, Ib. With this definition of the problem, an exact solution to the equations of motion was derived. The transverse position of bunch n, x,,, at location z is n-l X”(Z) = COS(Z&) + C Anj cos(z’BO) (-‘)j’ (3) j=l sin(z/p,) (-l)(j-1)n Presented at the Xvth International Conference on High Energy Accelerators (HEACC’92) Hamburg, Germany, July 20-24, 1992 where the upper expression is for j even, and the lower for j odd. Also, “-j-l j(j+2k-l)! An,j = c rk k!(i+k)! ’ (4) k=O f E WbbPoIb 2Eo and r E Wbb&b 4Eo . (5) Equation 3 shows that the growth of the bunch amplitudes is characterized by a power series in fz. Thus to keep the increase in the amplitudes small at the end of the linac, one