Propagation of discontinuities in an electromagnetic field (§3.1.1)

IT was mentioned in §3.1.1 that the eikonal equation of geometrical optics is identical with an equation which describes the propagation of discontinuities in an electromagnetic field. More generally, the four equations §3.1 (lla)-(14a) governing the behaviour of the electromagnetic field associated with the geometrical light rays may be shown to be identical with equations which connect the field vectors on a moving discontinuity surface. It is the purpose of this appendix to demonstrate this mathematical equivalence. Relations connecting discontinuous changes in field vectors In §1.1.3 we considered discontinuities in field vectors which arise from abrupt changes in the material parameters £ and fi, for example at a surface of a lens. Discontinuous fields may also arise from entirely different reasons, namely because a source suddenly begins to radiate. The field then spreads into the space surrounding the source and with increasing time fills a larger and larger region. On the boundary of this region the field has a discontinuity, the field vectors being in general finite inside this region and zero outside it. We shall first establish certain general relations which hold on any surface at which the field is discontinuous. For simplicity we assume that at any instant of time t > 0 there is only one such surface; the extension to several discontinuity surfaces (which may arise, for example, from reflections at obstacles present in the medium) is straightforward.