Fourier optics from the perspective of the Heisenberg group

Abstract

We discuss (linear) geometrical optics and its scalar wave counterpart, Fourier optics, as a natural outgrowth of the group theoretical machinery emanating from the symplectic structure of (optical) phase space, as it derives from the Fermat principle. We do, in fact, quantize geometrical optics. The symplectic structure is preserved by the (symplectic) group of linear canonical transformations. Every element of the symplectic group can be realized as a part of an image system in geometrical optics. On the other hand, one may associate a (nilpotent) Lie algebra to the (symplectic) phase space on which the symplectic group Sp (4, ℜ) acts in the natural way. Looking further on the irreducible unitary representations of the underlying simply connected Lie group, the Heisenberg group, one finds that the symplectic action on the Heisenberg group induces a group of unitary transformations in the space of each (infinite-dimensional) irreducible representation. This group, called the metaplectic group, describes the evolution of the (scalar) wave field (Fourier optics) corresponding to the evolution of rays (geometrical optics), and thereby connects the two views on optics. We put emphasis on the interplay of concepts, and for this reason, we also invoke intuition and terminology from classical and quantum mechanics.