ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics.

It is shown that every ray transfer matrix, often called the ABCD matrix, can be written as a similarity transformation of one of the Wigner matrices that dictate the internal space-time symmetries of relativistic particles, while the transformation matrix is a rotation preceded by a squeeze. The implementation of this mathematical procedure is described, and how it facilitates the calculations for scattering processes in periodic systems is explained. Multilayer optics and resonators such as laser cavities are discussed in detail. For both cases, the one-cycle transfer matrix is written as a similarity transformation of one of the Wigner matrices, rendering the computation of the ABCD matrix for an arbitrary number of cycles tractable.