The geometrical description of electromagnetic radiation

The paper describes the relationship between the solutions of Maxwell’s equations which can be considered at least locally as plane waves and the curvilinear coordinates of geometrical optics; it generalizes the results achieved by Lüneburg, concerning the evolution of surfaces of electromagnetic fields discontinuities. If vectors (Formula presented.) and (Formula presented.) are orthogonal to each other and their directions do not change with time t, but may vary from point to point in the domain G, then under some conditions there is an orthogonal coordinate system (Formula presented.) in which (Formula presented.) -lines represent rays of geometrical optics, (Formula presented.) -lines point out (Formula presented.) -direction and, (Formula presented.) -lines point out (Formula presented.) -direction. This coordinate system will be called phase-ray coordinate system. In the article, it will be proved that field under study can be represented by two scalar functions. The article will also specify the necessary and sufficient conditions for the existence of a coordinate system, generated by the solution of Maxwell’s equations with the holonomic field of the Poynting vector. It is shown that the class of solutions of Maxwell’s equations, as described in this work, includes monochromatic polarized waves, and the Hilbert–Courant solutions and their generalizations. © 2016 Informa UK Limited, trading as Taylor & Francis Group.