Self-Imaging Objects of Infinite Aperture*

TLDR

The paper derives necessary and sufficient conditions for an infinite plane object illuminated by a normal‑incidence monochromatic plane wave to self‑image without lenses or other optical accessories. The study investigates two cases of self‑imaging. By solving a reduced wave equation that violates the Sommerfeld radiation condition and applying a geometrical‑optics limit as λ→0, the authors establish conditions for weak self‑imaging (spatial frequencies ≪1/λ with the object’s Fourier spectrum on Fresnel‑zone‑plate circles) and strong self‑imaging (spatial frequencies up to 1/λ for doubly‑, singly‑periodic, and nonperiodic objects). For periodic objects, the derived conditions agree well with existing experimental and theoretical results based on the Fresnel–Kirchhoff diffraction integral with the parabolic approximation.

Abstract

The necessary and sufficient conditions are derived in order that an infinite plane object, illuminated by a plane monochromatic wave of normal incidence, images itself without the aid of lenses or other optical accessories. This involves a solution of the reduced wave equation which does not satisfy the Sommerfeld radiation condition. The solution is obtained by requiring a geometrical-optics limiting condition as the wavelength λ goes to zero. Two cases of self-imaging are considered. The first case, called weak, deals with the faithful imaging of objects whose spatial frequencies are all much smaller than the (1/λ) value of the illuminating source. The conditions for this case demand that the two-dimensional Fourier spectrum of the object lies on the circles of a Fresnel zone plate. The second case, called strong, deals with the faithful imaging of objects for spatial frequencies up to the natural cutoff of 1/λ. Both doubly- and singly-periodic and nonperiodic objects are considered. For periodic objects the results are shown to agree well with the experimental and theoretical work to date, the latter of which has always employed the Fresnel–Kirchhoff diffraction integral with the parabolic approximation appropriate to Fresnel diffraction.