Path–linking interpretation of Kirchhoff diffraction: a summary

Abstract

The Kirchhoff theory of diffraction of 1882 underpinned the earlier formulations of wave optics of Huygens, Young and Fresnel. Remarkably, it provides an exact explicit solution of the wave equation, with well–defined, though unusual, boundary conditions on the opaque diffracting screen. This prestigious status was not appreciated at the time and, despite the incisive clarification of Kottler in 1923, remains largely unsung to this day. Here I re–present my path–linking interpretation of Kirchhoff diffraction in which the Kirchhoff wave field at any observation position is formally represented as a ‘sum’ of contributions from all possible paths of propagation from the source point, whether or not they pass through the screen. The contributions are shown to be weighted according to the linking number of the path (when closed by an unobstructed return path) with the screen boundary loop(s). Kirchhoff optics is thus shown to be inherently topological in nature.