ABCD-systems with unimodular eigenvalues

It is shown that a lossless first-order optical system (or ABCD-system) whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues, is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the separable fractional Fourier transformer are related by means of a similarity transformation. Moreover, it is shown that the system that performs this similarity transformation, is itself a lossless first-order optical system