Sommerfeld-Runge Law in Three and Four Dimensions

The Sommerfeld-Runge law, a law of geometric optics expressing that the propagation vector is irrotational, is useful for general deductions and for computations of geometric-optical wave fields, for which it proves without supplementary notions sufficient under a wide range of conditions. Application to inhomogeneous anisotropic media, though not necessitating recourse to the concept of rays, yields the equation of ray paths. Absorption and lateral intensity variation of waves are accounted for by introducing complex propagation vectors.A four-dimensional generalization of the law is applicable to modulated waves and time-varying media. The four-dimensional considerations, to which it leads, deal with group propagation (in time-constant and time-varying media), with isotropy in space-time (relating to the law uv=c²), and with refraction and reflection at discontinuities in time and at moving boundaries. A brief discussion of focusing and diffraction in the scope of four-dimensional geometric optics is also given.