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A Sampling Theorem for Computational Diffraction

A major challenge of many diffraction calculations, using some form of the Rayleigh-Sommerfeld formulas, is the integration of a highly oscillatory integrand. Here we derive a potentially useful alternative form of solution to the Helmholtz equation, which implies a sampling theorem for the evaluation of a diffracted scalar field. This alternative solution bears close resemblance to the Rayleigh-Sommerfeld diffraction formulas, but instead incorporates the boundary conditions digitally. Hence, the integration is replaced by a simple summation. This formulation may be more efficient for accurate computer-based calculation of the diffracted scalar field.