A Sampling Theorem for Computational Diffraction

TL;DR

This paper introduces a sampling theorem-based method for diffraction calculations that replaces complex integrals with simple summations, potentially improving computational efficiency and accuracy.

Contribution

It derives a new solution to the Helmholtz equation that incorporates boundary conditions digitally, simplifying diffraction computations.

Findings

Replaces oscillatory integrals with summations

Potentially more efficient for computer calculations

Maintains accuracy in scalar field evaluation

Abstract

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.