BBA Procedure

The applied method [4,5] is based on the well known fact that if the strength of a single quadrupole $q$ in the ring is changed, the resulting difference in the closed orbit $\Delta y(s)$ is proportional to the original offset $y_q$ of the beam at $q$.
The equation for the resulting difference orbit is:

$$\Delta y''(s) - (k(s) + \Delta k(s))\Delta y(s) = \Delta k(s) y_q(s).$$

The difference orbit is thus given by the closed orbit formula for a single kick, but calculated with the perturbed optics including $\Delta k(s)$.

Figure 2: Illustration of the beam-based alignment technique applied to the quadrupoles with adjacent BPMs

From the measured difference orbit the kick and thus $y_q$ can be easily determined and compared to the nominal orbit $y_{bpm}$ in the BPM adjacent to the quadrupole, yielding the offset between BPM and quadrupole axis. The precision of the method is very much improved by taking difference orbit data for several local beam positions $y_q$ varied with an orbit bump. The principle of the method is illustrated in Fig. 2. The error of the nominal position $y_{bpm}$ for which the beam goes through the center of the quadrupole is then given by the resolution of the BPM system. In the SLS storage ring, a difference orbit with an amplitude of 5 $\mu$m can be clearly resolved [6]. This results in a resolution for the local kick of $\approx$ 0.25 $\mu$rad for quadrupoles at vertical beta values of 20 m. Since a change in quadrupole strength of $\Delta kl$ = 0.02 m$^{-1}$ causing a tune variation $\delta\nu$ = 0.03, is possible without losing the beam, a minimum beam offset of $y_{q,min}$ = 15 $\mu$m can be easily detected. Taking several data points by varying a local bump, the quadrupole-to-BPM alignment can be done with a precision of $\approx$ 5 $\mu$m. However some of the quadrupoles are at low beta values of 2.5 m which reduces the precision of the measurement to $\approx$ 40 $\mu$m.