Matrix perturbation approach to the weak linear coupling

The strict matrix treatment of linear coupling in the normalized coordinate system is presented. Then, assuming weak linear coupling, with the perturbed one-turn transfer matrix in the normalized coordinate system, the analytical expressions for the coupling parameters are obtained directly. Two complex coefficients h - and h + are defined to represent the weak linear difference and sum coupling, respectively. Numerical simulation was carried out to check these analytical expressions. The perturbed betatron motion of the weak linear coupling is discussed. Four eigenmode projection parameters are defined. Based on their measurement, the coupling parameters r and h - can be determined under weak linear difference coupling.