  The beam transfer matrix, often called the $ABCD$ matrix, is a two-by-two
matrix with unit determinant, and with three independent parameters. It is
noted that this matrix cannot always be diagonalized. It can however be brought
by rotation to a matrix with equal diagonal elements. This equi-diagonal matrix
can then be squeeze-transformed to a rotation, to a squeeze, or to one of the
two shear matrices. It is noted that these one-parameter matrices constitute
the basic elements of the Wigner's little group for space-time symmetries of
elementary particles. Thus every $ABCD$ matrix can be written as a similarity
transformation of one of the Wigner matrices, while the transformation matrix
is a rotation preceded by a squeeze. This mathematical property enables us to
compute scattering processes in periodic systems. Laser cavities and multilayer
optics are discussed in detail. For both cases, it is shown possible to write
the one-cycle transfer matrix as a similarity transformation of one of the
Wigner matrices. It is thus possible to calculate the $ABCD$ matrix for an
arbitrary number of cycles.
