A Geometrical Perspective on the Coherent Multimode Optical Field and Mode Coupling Equations

The generalization of the Poincaré sphere to N ≥ 2 modes is the (N - 1)-dimensional complex projective space CP(N - 1). There is a minimal set of 2N - 2 Stokes vector components that determine the coherent multimode optical field. These are obtained from the inverse stereographic projection of coordinate hyperplanes in CP(N -1) into a 2N -2 sphere, just as in the N = 2 case. We derive N-mode analogs of Poole's optical fiber polarization-mode dispersion (PMD) equations that involve only 2N - 2 independent variables. This is achieved by means of an explicit generalized coherent state representation of the optical field, which enables the components of the PMD vector to be expressed in terms of the optical state and its frequency derivatives. Poole's equations describe mode coupling as a flow on CP(N -1). We give general constraints on the mode-coupling matrix and Stokes vector components. The group delay operator is shown to be a rank-2 perturbation of a diagonal matrix.