Proposed Implementation of “Non-Physical” Four-Dimensional Polarization Rotations

Recently one of us proposed a new formalism for modeling electromagnetic wave transformations for coherent communication using a real, four-vector description instead of the conventionally used Jones calculus or the Mueller matrices. The four-vector can then handle all superpositions of two orthogonal polarization basis and two orthogonal time bases (e.g., the in-phase and quadrature phase). In developing this formulation it was found that to provide a general but minimal framework for such rotations, it is natural to divide the six generators of four-dimensional (4d) rotations into two groups of three generators, the right- and the left-isoclinic matrices. Of the six transformations these generators define, it was furthermore found that four of them are readily implemented by linear optical components, while two of then were impossible to implement by such means. In this paper, we detail the reason these two “unphysical” rotations cannot be implemented with linear optics. We also suggest how they can be implemented, but at a cost in the signal-to-noise ratio, and give this minimum cost.