Analytical boundary-based method for diffraction calculations

Abstract

We present a simple method for calculation of diffraction effects in a beam passing an aperture. It follows the wellknown approach of Miyamoto and Wolf, but is simpler and does not lead to singularities. It is thus shown that in the near-field region, i.e., at short propagation distances, most results depend on values of the beam's field at the aperture's boundaries, making it possible to derive diffraction effects in the form of a simple contour integral over the boundaries. For a uniform, i.e., plane-wave incident beam, the contour integral predicts the diffraction effects exactly. Comparisons of the analytical method and full numerical solutions demonstrate highly accurate agreement between them.

Renewed diffraction formula using reciprocity

“…Subsequently, Kirchhoff further solidified Fresnel's formula by deriving it from the integral solution to the Helmholtz equation [1]. Due to the mathematical rigor of Kirchhoff's integral theorem, this foundational concept has been widely utilized in optics to develop theoretical models for understanding the diffraction patterns produced by slits, gratings, and edges [2][3][4][5][6][7][8][9][10]. Additionally, this theory has found applications in other domains.…”