There are three goals in the accurate nonlinear diagnosis of a storage ring. First, the beam must be moved to amplitudes many times the natural beam size. Second, strong and long lasting signals must be generated. Third, the measurement technique should be non-destructive.Conventionally, a single turn kick moves the beam to large amplitudes, and turn-by-turn data are recorded from multiple beam position monitors (BPMs) [1][2][3][4][5][6]. Unfortunately, tune spread across the beam causes the center of charge beam signal to "decohere" on a time scale often less than 100 turns. Filamentation also permanently destroys the beam emittance (in a hadron ring). Thus, the "strong single turn kick" technique successfully achieves only one out of the three goals. AC dipole techniques can achieve all three. Adiabatically excited AC dipoles slowly move the beam out to large amplitudes. The coherent signals then recorded last arbitrarily long. The beam maintains its original emittance if the AC dipoles are also turned off adiabatically, ready for further use.The AGS already uses an RF dipole to accelerate polarized proton beams through depolarizing resonances with minimal polarization loss [7]. Similar AC dipoles will be installed in the horizontal and vertical planes of both rings in RHIC [8]. The RHIC AC dipoles will also be used as spin flippers, and to measure linear optical functions [9].
LINEAR MOTIONHorizontal motion is described using complex phasorsso that the unperturbed one turn motion is justwhere R = exp(i 2πQ X ). Here Q X is the betatron tune, and the normalized coordinates x and x both have the dimensions of length. An AC dipole just after the reference point gives a real normalized angular kick on turn t ofwhere Q D is the drive tune and ψ 0 is the initial phase. The AC dipole strength is δ = (BL/(Bρ)) β D , where BL is the integrated field amplitude, (Bρ) is the rigidity, and β D is the Twiss function at the dipole. If z = z 0 just before the first dipole kick, then the net displacement phasor on turn T is The exact general solution for linear motion is [10] (5) where z = z 0 − δ − + δ + is a constant given by the initial conditions, and the complex AC dipole strengths arewhereThe oscillating closed orbit is defined as that orbit which exactly repeats itself after one modulation period. The solution on turn T is obtained by putting z = 0, so thatgenerally following a tilted ellipse, not a circle, in normalized phase space. The semi-minor and semi-major axes are ||δ − | − |δ + || and |δ − | + |δ + | long. In practice the aspect ratio of the ellipse is close to 1 when the AC dipole is driven at a tune close to the fractional betatron tune (Q − ≈ 0).Motion in the rotating frame, which rotates with the AC dipole drive at 2πQ D per turn, is denoted by over-bars. Assuming the previous approximation to be accurate,That is, a test particle slowly circulates the vector δ − at a radius of constant length | z |, as illustrated in Fig. 1. When a bunch is considered, a distribution of z values must be used. A smoothly di...
