where E0 is the slowly varying amplitude and the phase Ψ is a function of the position vector q and time t. If the field is observed only at locations that are very far from any source, the phase can be expressed as

where k is the wave vector, ω is the angular frequency, and α is an arbitrary phase shift. In an isotropic medium, the wave vector and the angular frequency are further related by the expression

where n(q) is the refractive index of the medium. It follows that the wave vector and angular frequency can be expressed in terms of the phase:

Following Landau and Lifshitz in Ref. 1, the wave vector and frequency are analogous to the generalized momentum p and Hamiltonian H of a solid particle,

where S is the integral of the Lagrangian along the particle’s trajectory. From this analogy, it follows that the ray trajectory can be computed by solving six coupled first-order ordinary differential equations for the components of k and q: