Integral-Transform Formulation of Diffraction Theory*

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Abstract
An exact integral-transform formulation of the theory of the diffraction of monochromatic scalar waves from an infinite plane boundary with known boundary values is developed from first principles. The formulation is analogous to the Fourier-transform formulation of Fraunhofer diffraction, except that it is exact and is valid for both Fresnel and Fraunhofer diffraction. The known wavefunction on the boundary is expanded into a linear superposition of point-convergence patterns. The point-convergence pattern is defined as the pattern of the wavefunction that must exist on the boundary in order to produce a Dirac delta-function distribution of the wavefunction on a plane of observation that is parallel to the boundary. This pattern is described by a point-convergence function h determined by the position of the delta-function singularity on the plane of observation and the distance of that plane from the boundary. The point-convergence function is orthogonal to a function g called the point-divergence function. Together, g and h serve as kernels for a new integral transform called the diffraction transform. The pattern of the wavefunction on the plane of observation is the diffraction transform of the wavefunction on the boundary. We show that g represents the field of a dipole while h cannot be expressed by an ordinary function and must be defined as a functional. Successive diffraction transformations are discussed, and the transform formulation of Fraunhofer diffraction is obtained as a special case from the diffraction-transform theory.

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