Parametrization of Linear Coupled Motion in Periodic Systems

The Courant-Snyder parametrization for one-dimensional linear motion in periodic systems is generalized to two-dimensional coupled linear motions. The 10-parameter 4×4 symplectic transfer matrix across a period is expressed in two normal-mode invariant phase-advances, four normal-mode periodic "amplitude" functions, and four periodic functions which reflect the strength and structure of the coupling. These parametric functions are also given by differential equations containing the periodic force coefficients appearing in the Hamiltonian. Two bilinear invariants can be constructed. With given horizontal and vertical aperture limitations, these invariants give the four-dimensional phase-space acceptance volume and the propagation of the horizontal and vertical emittances of the beam.