An attempt was made to measure the betatron coupling with 'Mille-Tour' BPMs (MTBPM). Owing to its capability of averaging reproducible trajectories, MTBPM provides highly accurate turn-by-turn data. The scheme developed makes full use of this, as follows. For each straight section that has no focusing element between two BPMs: 1) Construct the phase space (x, x', z, z'); 2) Fit the phase space data to extract a 4x4 one turn matrix; 3) Perform normal mode decomposition to obtain the normal modes as well as the rotation matrix that transforms the geometric modes to the former. Step 2 is found to be the most non-trivial, especially as the number of available turns is severely limited due to the strong decoherence of the beam. To minimise the decoherence, measurements were made with a low beam current in 1/3 filling, a small oscillation amplitude given by an injection kicker, and a sextupole setting that gives zero chromaticities and minimal tune shift with amplitude.

The validity of the results can be readily seen in the decoupling of the two normal modes (Figure 128), which worked successfully over a wide range of coupling, down to 17 pm.rad vertical emittance (~0.5% coupling) measured with a pinhole camera. Although the ratio of the two normal mode ellipses closely followed the measured coupling in most cases, it strictly depends on the initial condition. The magnitude of elements in the rotation matrix, instead, represents the local coupling of the machine, and could therefore be utilised for modelling as well as correction of the coupling. 

Figure 128
Fig. 128: An example of the decomposition, showing the transformation from the vertical phase space (blue) to normal mode (red), obtained for the standard operation setting (~30 pm.rad vertical emittance).

To prepare for the 5 GeV operation performed in October 2000 in the low coupling mode, the modelling of the coupling was made at 5 GeV with a response matrix measurement. Applying the corrector strengths obtained from the model gave immediately 9~10 pm.rad on the ID25 pinhole, indicating that the modelling works over a wide range of machine settings. Further coupling reduction with the empirical minimisation of the pinhole image depended greatly on the vertical stability of the beam. Thanks to a quiet beam achieved at 200 mA in 2x1/3 filling with RF modulation, free of both resistive-wall and ion instabilities, values as low as 5 pm.rad were measured on the ID25 pinhole, hitting the resolution limit of the device. Using the obtained setting, and with the automatic coupling correction loop, the coupling could be kept around 6~7 pm.rad level during the 5 GeV operation.