Alternative description of the linear canonical integral transformation

First-optical systems (or ABCD-systems) with a singular submatrix B are considered. Starting with the Iwasawa-type decomposition of a first-order optical system as a cascade of a lens, a magnifier, and an ortho-symplectic system (a system that is both symplectic and orthogonal), a further decomposition of the ortho-symplectic system in the form of a separable fractional Fourier transformer embedded in between two spatial-coordinate rotators is proposed. The resulting decomposition of the entire firstorder optical system leads to a representation of the linear canonical integral transformation, which is valid also in the case of a singular submatrix B. Some examples of ABCDsystems with a singular submatrix B are given.