Generalization of the Angular Spectrum of Plane Waves and the Diffraction Transform*

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Abstract
The generalized angular spectrum of plane waves yields scalar Helmholtz-equation solutions that satisfy prescribed boundary values on an infinite plane boundary when the boundary values are described by ultradistributions [generalized functions in the space denoted (z′)]. These solutions are ultradistributions that satisfy the Helmholtz equation in all space. The generalized angular spectrum permits the treatment of boundary values that cannot be treated by classical diffraction formulas. The utility of the generalized angular spectrum is demonstrated by treatment of (1) boundary values described by absolutely integrable or square-integrable functions, (2) examples of boundary values described by functions that do not have Fourier transforms in the classical sense, and (3) examples of boundary values described by ultradistributions that are not expressible as functions. The generalized angular spectrum is used to develop, within the framework of established generalized-function theory, the transform formulation of diffraction of Sherman and the inverse-diffraction problem of Wolf and Shewell. A concise summary is given in the Appendix of the basic concepts and formulas of generalized-function theory that are useful in applied analysis.