A Note on the Backward Scattering Theorem
Recent efforts by C.-T. Tai to emphasize backward scattering within the makeup of the optical theorem are examined here from first principles. The present work exploits spectral field representations and a common asymptotic procedure so as to build up both the scattered fields and their contribution to the extinction integral. The result of all this is to reaffirm the strictly forward scattering nature of the optical theorem as commonly understood, while at the same time reconciling it with a backward scattering interpretation. The backward scattering, it so turns out, is backward in reciprocal space, wherein it affects the Fourier transform of the currents induced throughout the scattering object. The standard forward scattering attribute of the optical theorem, forward in the context of actual space, remains unimpaired. In truth, however, the backward spectral attribute is a mere technical formality, made available for only one of the two signature options which one can exercise when making specific the details of transformation. The alternate signature option leads to a forward appearance in spectral space also, with the actual value of the current transform appearing in the optical theorem quite intact. We develop these results in detail and then, for completeness, summarize the special form which they adopt for scattering obstacles with axial symmetry.