Reducing canonical diffraction problems to singularity-free one-dimensional integrals

Abstract

The oscillatory integrands of the Kirchhoff and the Rayleigh–Sommerfeld diffraction solutions mean that these two-dimensional integrals typically lead to challenging computations. By adoption of the Kirchhoff boundary conditions, the domain of the integrals is reduced to cover only the aperture. For perfect spherical (both diverging and focused) and plane incident fields, closed forms are derived for vector potentials that allow each of these solutions to be further simplified to just a one-dimensional, singularity-free integral around the aperture rim. The results offer easy numerical access to exact—although, given the approximate boundary conditions, not rigorous—solutions to important diffraction problems. They are derived by generalization of a standard theorem to extend previous results to the case of focused fields and the Rayleigh–Sommerfeld solutions.