 Fresnel half-period zone

A Fresnel half-period zone is a region of an aperture whose optical path to an observation point differs from the axial path by an interval of half a wavelength. For an axial optical point source a distance $s$ before the aperture and an axial observer a distance $r$ beyond it, the paraxial path excess at aperture radius $u$ is $u^2(s^{-1}+r^{-1})/2$. The outer radius of zone $n$ is thereforeSuccessive zones contribute approximately opposite phases to the Huygens-Fresnel principle integral. The finite source distance matters: using the plane-wave formula $u_n^2=n\lambda r$ requires $s\to \infty$.