22 - Geometric-optical rays, Poynting's vector, and the field momenta

Summary

In isotropic media the rays of geometrical optics are usually obtained from the surfaces of constant phase (i.e., wavefronts) by drawing normals to these surfaces at various points of interest. It is also possible to find the rays from the eikonal equation, which is derived from Maxwell's equations in the limit when the wavelength λ of the light is vanishingly small. Both methods provide a fairly accurate picture of beam-propagation and electromagnetic-energy transport in situations where the concepts of geometrical optics and ray-tracing are applicable. The artifact of rays, however, breaks down near caustics and focal points and in the vicinity of sharp boundaries, where diffraction effects and the vectorial nature of the field can no longer be ignored.

It is possible, however, to define the rays in a rigorous manner (consistent with Maxwell's electromagnetic theory) such that they remain meaningful even in those regimes where the notions of geometrical optics break down. Admittedly, in such regimes the rays are no longer useful for ray-tracing; for instance, the light rays no longer propagate along straight lines even in free space. However, the rays continue to be useful as they convey information about the magnitude and direction of the energy flow, the linear momentum of the field (which is the source of radiation pressure), and the angular momentum of the field. Such properties of light are currently of great practical interest, for example, in developing optical tweezers, where focused laser beams control the movements of small objects.