Transverse beam sizes and quasi emittances for linearly coupled optics

Two transverse eigenmode rms emittances are defined statistically in the uncoupled normalized coordinate system. They are assumed to be adiabatic invariants. With linear coupling's action-angle parameterization, the strict expression for the beam size matrix in the laboratory coordinate system is obtained. It can be expressed in matrix P defined in the action-angle parameterization or in Twiss and coupling parameters defined in Edwards-Teng's parameterization, along with the eigenmode rms emittances. Numerical simulation calculations are carried out to check the analytical expressions and to verify the adiabatic eigenmode rms emittance invariants during the skew quadrupole magnet ramping. With linear coupling's matrix perturbation approach, the strict expressions of the horizontal and vertical beam sizes in the laboratory coordinate system are approximated to give the quasi horizontal and vertical beam emittances under the weak linear difference coupling situation. The predictions of quasi horizontal and vertical beam emittances from different approximations are presented and discussed.