Linear coupling parametrization in the action-angle frame

Abstract

The linear coupling's parameterization in the uncoupled action-angle frame is introduced, where the two eigenmodes' phases at one point and the phase advances between two points in the ring are well defined. The general expressions of the coordinates $(x,{x}^{\ensuremath{'}},y,{y}^{\ensuremath{'}})$ are also given. The transfer matrix $\mathbf{P}$ from the action-angle frame to the laboratory frame can be obtained from the eigenvectors of the one-turn transfer map $\mathbf{T}$ or from Twiss and coupling parameters defined in Edwards-Teng parameterization. Matrix $\mathbf{P}$ can also be constructed from turn-by-turn beam position monitor (BPM) data. The phase ellipses and beam sizes in different projection planes are calculated through the action-angle parameterization. The two eigenmodes' Twiss and coupling parameters at point ${s}_{2}$ are described in terms of the section transfer map ${\mathbf{T}}_{1\ensuremath{\rightarrow}2}$ and parameters at point ${s}_{1}$. The linear coupling's action-angle parameterization is useful for the analytical calculations and turn-by-turn BPM data interpretations.