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

William A. Wood, Willard Miller, Michal Mlejnek

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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.

Original languageEnglish (US)
Article number7115021
JournalIEEE Journal of Quantum Electronics
Issue number7
StatePublished - Jul 1 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 1965-2012 IEEE.


  • Poincare sphere
  • complex projective space
  • mode coupling
  • multimode optical fiber


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