Discrete Hopf fibration in the design of four-dimensional modulations

In this work we demonstrate how the formalism of Hopf fibration can be useful in the context of classical coherent optical communications. We have found that the Hopf maps can provide complementary information on the geometry of four-dimensional signal space through explicit and concise relations. Using the sampled discrete Hopf fibration, we designed a dense 4D modulation called 14PolSK-8QAM and for practical purposes we demonstrate the close relationship between the Hopf maps and the transfer function of the Polarization Multiplexed Quadrature Mach-Zehnder modulators.