Variational formulation of covariant eikonal theory for vector waves

Abstract

Abstract The eikonal theory of wave propagation is developed by means of a Lorentz-covariant variational principle, involving functions defined on the natural eight-dimensional phase space of rays. The wave field is a four-vector representing the electromagnetic potential, while the medium is represented by an anisotropic, dispersive nonuniform dielectric tensor D μν ( k , x ). The eikonal expansion yields, to lowest order, the hamiltonian ray equations, which define the lagrangian manifold k ( x ), and the wave-action conservation law, which determines the wave-amplitude transport along the rays. The first-order contribution to the variational principle yields a concise expression for the transport of the polarization phase. The symmetry between k -space and x -space allows for a simple implementation of the Maslov transform, which avoids the difficulties of caustic singularities.