Group theoretical methods in optics

Scalar Fourier optics is concerned with the passage of paraxial light beams through ideal optical systems. It is well known that the action of the latter on the former can be given in the framework of the two- and four-dimensional real symplectic groups. It is shown here that, based on an analysis of the Poincaré symmetry of the complete Maxwell equations in the front form, a natural representation for paraxial Maxwell beams emerges, which moreover shows the way to a generalization of scalar to vector Fourier optics preserving the group structure of ideal optical systems. Properties of generalized rays, and the usefulness of some pseudo-orthogonal groups in the treatment of Gaussian Schell-model beams, are also brought out.