Parametrization of the driven betatron oscillation

An AC dipole is a magnet which produces a sinusoidally oscillating dipole field and excites coherent transverse beam oscillations in a synchrotron. By observing this driven coherent oscillation, the linear optical parameters can be directly measured at locations of the beam position monitors. The driven oscillations induced by an AC dipole will generate a phase space ellipse which differs from that of free oscillations. If not properly accounted for, this difference can lead to misinterpretations of the actual optical parameters, for instance, 6% or more in the cases of the Tevatron, RHIC, or LHC. This paper shows that the effect of an AC dipole on the observed linear optics is identical to that of a thin lens quadrupole. By introducing a new amplitude function to describe this new phase space ellipse, the motion produced by an AC dipole becomes easier to interpret. The introduction of this new amplitude function also helps measurements of the normal Courant-Snyder parameters based on beam position data taken under the influence of an AC dipole. This new parametrization of driven oscillations is presented and is used to interpret data taken in the FNAL Tevatron using an AC dipole.

Figure 1

Schematic of incoherent free oscillations (gray) and excited coherent oscillations (black) of charged particles in the Tevatron. Because individual particles within the beam oscillate incoherently, coherent oscillations must be excited to observe the betatron oscillations and measure parameters such as the $\beta$ function. An AC dipole is one of such tools to excite coherent oscillations of the beam.

Figure 2

The amplitudes of the two modes driven by an AC dipole. A circulating beam is driven by the two driving terms of an AC dipole whose driving tunes are ${\nu }_d$ and $1-{\nu }_d$. Solid and dashed curves represent the amplitudes of the two modes excited by these two driving terms for a given machine tune. When the machine tune is $\nu$, beam motion is dominated by the mode of ${\nu }_d$ (solid curve). However, the mode of $1-{\nu }_d$ (dashed curve) cannot always be ignored since realizable ${\delta }_d={\nu }_d-\nu$ is limited by the tune spread of the beam (shaded area).

Figure 3

Ratio between the amplitude functions of driven and free oscillations, ${\beta }_d(s)$ and $\beta (s)$. Based on Eq. (6), ${\beta }_d(s)/\beta (s)$ is calculated for different values of ${\delta }_d={\nu }_d-\nu$ and $\psi (s)$. When $\nu =0.58$ like the Tevatron, compared to when $\nu =0.3$ like the RHIC and LHC, ${\beta }_d(s)/\beta (s)$ deviates larger from the unity and has stronger nonlinearity because of the larger influence of the secondary driving term.

Figure 4

The amplitude functions of free and driven oscillations, $\beta (s)$ (solid) and ${\beta }_d(s)$ when ${\delta }_d=-0.01$ (dashed), in the Tevatron. As expected, ${\beta }_d(s)$ shows beating with 10%–15% peak height relative to $\beta (s)$. From multiple data sets of driven oscillations, the amplitude function of free oscillations, $\beta (s)$, can be extrapolated.

Figure 5

Phase space ellipses of driven oscillations at an interaction point in the Tevatron (B0) when ${\delta }_d=\pm 0.02$ and $\pm 0.04$. Because the Courant-Snyder-like parameters of driven oscillations ${\beta }_d(s)$, ${\alpha }_d(s)$, and ${\gamma }_d(s)$ depend on ${\delta }_d$, not only the areas but also the shapes of the ellipses are different.

Figure 6

${\beta }_d(s)$ vs ${\delta }_d$ at an interaction of the Tevatron (B0). Each data point is from the fit of Eq. (24) to an ellipse in Fig. 5. The curve is the fit of Eq. (6) to the data points. The true $\beta$ function of this location is the value of ${\beta }_d(s)$ when ${\delta }_d=0$.

Figure 7

The amplitude of driven oscillations vs the primary driving tune ${\nu }_d$ at three BPM locations in the Tevatron. Two curves represent the fits when the secondary driving term is included (solid) and not included (dashed). The asymmetries around the machine tune $\nu \simeq 0.5785$ depend on the factor $\cos [2\psi (s)-2\pi \nu ]$.

Figure 8

The machine tune derived from the fit of the amplitude response at each BPM location. Inclusion of the secondary driving term gives a much more consistent result for the machine tune (global parameter).