Plane-wave Expansions of the Optical Field*

In the theories of electrodynamics, acoustics, and optics, fields that satisfy the homogeneous wave equation are often represented by integral expansions of monochromatic plane waves, all propagating with the same speed [ 1,2] . Two different plane-wave expansions are in use. One expansion, due to Whittaker [3] , expresses the field as a superposition of monochromatic, homogeneous plane waves; the other one, first employed by Weyl [4] to represent a spherical wave, consists of a superposition of monochromatic, homogeneous and inhomogeneous plane waves, Although both types of expansion have been used extensively since the work of Whittaker and Weyl, there are still important, unanswered questions concerning them. A systematic method for obtaining the amplitudes of the plane waves in the expansions is not always available, the domains of validity of the expansions are not fully known, and the relationship between the two different expansions for the same field, in space-time regions where both are valid, is not yet understood. In this paper, we provide the answers to these questions, for wave fields radiated by sources confined to a finite space-time domain **. The details of our analysis, a more complete discussion of our results, and an examination of other related questions will be presented elsewhere in an expository paper [6] . Let us consider the scalar wave field? V(r, t) radiated by a source described by a known, real, continuous function p(r, t) with continuous first partial derivatives, which is nonzero only in a bounded region of space-time, viz.