Dirac-bracket structure in multidimensional mode conversion

The intersection of two (2n − 1)-dimensional dispersion manifolds D a and D b in the 2n-dimensional ray phase space P yields a (2n − 2)-dimensional conversion manifold M ≡ D a ∩ D b that naturally possesses a Dirac-bracket structure that is inherited from the canonical Poisson bracket on ray phase space. The canonical symplectic two-form Ω ≡ Ω∥ + Ω⊥, defined on the 2n-dimensional tangent plane T z 0 P ≡ T z 0 M ⊕ ( T z 0 M ) ⊥ , can thus be decomposed into the Dirac two-form Ω∥ on the (2n − 2)-dimensional tangent plane T z 0 M at a conversion point z 0 ∈ M , and the symplectic two-form Ω⊥ on its orthogonal 2-dimensional complement ( T z 0 M ) ⊥ . These two symplectic two-forms are introduced in our analysis of multidimensional mode conversion, where their respective geometrical roles are defined. We note that since the Dirac-bracket structure Ω∥ vanishes identically when n = 1, it represents a new structure in multidimensional (n > 1) mode conversion theory.