Multiple resonance compensation for betatron coupling and its equivalence with matrix method

Analyses of betatron coupling can be broadly divided into two categories: the matrix approach that decouples the single-turn matrix to reveal the normal modes and the hamiltonian approach that evaluates the coupling in terms of the action of resonances in perturbation theory. The latter is often regarded as being less exact but good for physical insight. The common opinion is that the correction of the two closest sum and difference resonances to the working point is sufficient to reduce the off-axis terms in the 4X4 single-turn matrix, but this is only partially true. The reason for this is explained, and a method is developed that sums to infinity all coupling resonances and, in this way, obtains results equivalent to the matrix approach. The two approaches is discussed with reference to the dynamic aperture. Finally, the extension of the summation method to resonances of all orders is outlined and the relative importance of a single resonance compared to all resonances of a given order is analytically described as a function of the working point.